Scalable topological data analysis

ABSTRACT

An example method comprises receiving first data associated with data points, receiving a lens function selection, a metric function selection, and a resolution function, the metric function identified by the metric function selection being capable of performing functions on data as matrix functions, mapping second data based on the first data to a reference space by utilizing matrix vector multiplication for application of selected lens function on second data based on the first data to map the second data to the reference space, generating cover of reference space including the second data, clustering second data in cover based on the selected metric function to determine each node of a plurality of nodes, each of the nodes of the plurality of nodes comprising members representative of at least one subset of the data points, and generating a visualization comprising the plurality of nodes and a plurality of edges wherein each of the edges of the plurality of edges connects nodes with shared members.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Patent Application Ser. No.62/357,922, filed Jul. 1,2016, entitled “Systems and Methods forMetric/Lens Computations for Topological Analysis,” which isincorporated by reference.

BACKGROUND

1. Field of the Invention(s)

Embodiments discussed herein are directed to topological data analysisusing at least one function representable as a matrix function and moreparticularly, utilizing metric matrix functions for clustering datapoints projected in a covered reference space for scale topologicalanalysis.

2. Related Art

As the collection and storage of data has increased, there is anincreased need to analyze and make sense of large amounts of data.Examples of large datasets may be found in financial services companies,oil exploration, insurance, health care, biotech, and academia.Unfortunately, previous methods of analysis of large multidimensionaldatasets tend to be insufficient (if possible at all) to identifyimportant relationships and may be computationally inefficient.

In order to process large datasets, some previous methods of analysisuse clustering. Clustering often breaks important relationships and isoften too blunt an instrument to assist in the identification ofimportant relationships in the data. Similarly, previous methods oflinear regression, projection pursuit, principal component analysis, andmultidimensional scaling often do not reveal important relationships.Further, existing linear algebraic and analytic methods are toosensitive to large scale distances and, as a result, lose detail.

Even if the data is analyzed, sophisticated experts are often necessaryto interpret and understand the output of previous methods. Althoughsome previous methods allow graphs that depict some relationships in thedata, the graphs are not interactive and require considerable time for ateam of such experts to understand the relationships. Further, theoutput of previous methods does not allow for exploratory data analysiswhere the analysis can be quickly modified to discover newrelationships. Rather, previous methods require the formulation of ahypothesis before testing.

SUMMARY OF THE INVENTION(S)

An example method comprises receiving first data associated with datapoints, receiving a lens function selection, a metric functionselection, and a resolution function, the metric function identified bythe metric function selection being capable of performing functions ondata as matrix functions, mapping second data based on the first data toa reference space by utilizing matrix vector multiplication forapplication of the selected lens function on second data based on thefirst data to map the second data to the reference space, generatingcover of reference space including the second data (e.g., the coverbeing generated based on, at least in part, the resolution function),clustering second data in cover based on the selected metric function todetermine each node of a plurality of nodes, each of the nodes of theplurality of nodes comprising members representative of at least onesubset of the data points, and generating a visualization comprising theplurality of nodes and a plurality of edges wherein each of the edges ofthe plurality of edges connects nodes with shared members.

In various embodiments, the metric function is a cosine distancefunction, a correlation function, a Hamming function, or an L₂ metricfunction. The cosine distance function being representable in a matrixform:D _(cos)=11^(T)−diag(diag(XX^(T)))^(−1/2)XX^(T)diag(diag(XX^(T)))^(−1/2)where again X is a matrix that stores a dataset, “1” is a vector of allones and “1^(T)” is a transpose of the vectors of all ones. The acorrelation metric function being representable in a matrix form:D=11^(T)−diag(diag(YY^(T))^(−1/2)YY^(T)diag(diag(YY^(T)))^(−1/2)where Y=X−μ(X)1^(T). Here, μ(V) is the mean value of the entries of thevector V. The Hamming metric function being representable in a matrixform:D _(hamming) =X(11^(T) −X ^(T))The L₂ metric function being representable in a matrix form:D∘D=−2XX^(T)+diag(XX^(T))1^(T)+1diag(XX^(T))^(T)

In various embodiments, the lens function is an L₁ centrality function,an L₂ centrality function, a gaussian density function, a metric PCAfunction, or an MDS function. If the lens function is the L₁ centrality,given a distance operator D based on the metric function, the L₁centrality of each row in a data matrix is encoded in a vector D·1. Ifthe lens function is an L₂ centrality function and a distance operator Dis based on the metric function, the L₂ centrality of each row in a datamatrix is encoded in an entry-wise square root of a vector D∘D·1,wherein D∘D is a Hadamard square. If the lens function is a gaussiandensity function and a distance operator D is based on the metricfunction, an entry-wise function exp(−(·)) is performed on a Hadamardsquare of the distance operator to each entry. If the lens function is ametric PCA function, the lens function may utilize a distance operator Dbased on the metric function. If the lens function is an MDS function,the lens function may utilize a Hadamard square of the distanceoperator.

An example non-transitory computer readable medium comprisinginstructions executable by a processor to perform a method. The methodmay comprise receiving first data associated with data points, receivinga lens function selection, a metric function selection, and a resolutionfunction, the metric function identified by the metric functionselection being capable of performing functions on data as matrixfunctions, mapping second data based on the first data to a referencespace by utilizing matrix vector multiplication for application ofselected lens function on second data based on the first data to map thesecond data to the reference space, generating cover of reference spaceincluding the second data, clustering second data in cover based on theselected metric function to determine each node of a plurality of nodes,each of the nodes of the plurality of nodes comprising membersrepresentative of at least one subset of the data points, and generatinga visualization comprising the plurality of nodes and a plurality ofedges wherein each of the edges of the plurality of edges connects nodeswith shared members.

An example system comprises one or more processors and memory containinginstructions executable by the processor to: receive first dataassociated with data points, receive a lens function selection, a metricfunction selection, and a resolution function, the metric functionidentified by the metric function selection being capable of performingfunctions on data as matrix functions, map second data based on thefirst data to a reference space by utilizing matrix vectormultiplication for application of selected lens function on second databased on the first data to map the second data to the reference space,generate cover of reference space including the second data, clustersecond data in cover based on the selected metric function to determineeach node of a plurality of nodes, each of the nodes of the plurality ofnodes comprising members representative of at least one subset of thedata points, and generate a visualization comprising the plurality ofnodes and a plurality of edges wherein each of the edges of theplurality of edges connects nodes with shared members.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is an example graph representing data that appears to be dividedinto three disconnected groups.

FIG. 1B is an example graph representing data set obtained from aLotka-Volterra equation modeling the populations of predators and preyover lime.

FIG. 1C is an example graph of data sets whereby the data does not breakup into disconnected groups, but instead has a structure in which thereare lines (or flares) emanating from a central group.

FIG. 2 is an example environment in which embodiments may be practiced.

FIG. 3 is a block diagram of an example analysis server.

FIG. 4 is a flow chart depicting an example method of dataset analysisand visualization in some embodiments.

FIG. 5 is an example ID field selection interface window in someembodiments.

FIG. 6a is an example data field selection interface window in someembodiments.

FIG. 6b is an example metric and filter selection interface window insome embodiments.

FIG. 7 is an example filter parameter interface window in someembodiments.

FIG. 8 is a flowchart for data analysis and generating a visualizationin some embodiments.

FIG. 9 is an example interactive visualization in some embodiments.

FIG. 10 is an example interactive visualization displaying an explaininformation window in some embodiments.

FIG. 11 is a flowchart of functionality of the interactive visualizationin some embodiments.

FIG. 12 is a flowchart of for generating a cancer map visualizationutilizing biological data of a plurality of patients in someembodiments.

FIG. 13 is an example data structure including biological data for anumber of patients that may be used to generate the cancer mapvisualization in some embodiments.

FIG. 14 is an example visualization displaying the cancer map in someembodiments.

FIG. 15 is a flowchart of for positioning new patient data relative tothe cancer map visualization in some embodiments.

FIG. 16 is an example visualization displaying the cancer map includingpositions for three new cancer patients in some embodiments.

FIG. 17 is a flowchart of utilization the visualization and positioningof new patient data in some embodiments

FIG. 18 is an example digital device in some embodiments.

FIG. 19 is a flow chart for using at least one function representable asa matrix function and more particularly, utilizing metric matrixfunctions for clustering data points projected in a covered referencespace for scale topological analysis.

FIG. 20 is a graph of an example of computing L₁ centrality over a dataset with 100 columns and a variable number of rows using vectormultiplication as discussed in some embodiments described herein

DETAILED DESCRIPTION OF DRAWINGS

Some embodiments described herein may be a part of the subject ofTopological Data Analysis (TDA). TDA is an area of research which hasproduced methods for studying point cloud data sets from a geometricpoint of view. Other data analysis techniques use “approximation bymodels” of various types. Examples of other data analysis techniquesinclude regression methods which model data as a graph of a function inone or more variables. Unfortunately, certain qualitative properties(which one can readily observe when the data is two-dimensional) may beof a great deal of importance for understanding, and these features maynot be readily represented within such models.

FIG. 1A is an example graph representing data that appears to be dividedinto three disconnected groups. In this example, the data for this graphmay be associated with various physical characteristics related todifferent population groups or biomedical data related to differentforms of a disease. Seeing that the data breaks into groups in thisfashion can give insight into the data, once one understands whatcharacterizes the groups.

FIG. 1B is an example graph representing data set obtained from aLotka-Volterra equation modeling the populations of predators and preyover time. From FIG. 1B, one observation about this data is that it isarranged in a loop. The loop is not exactly circular, but it istopologically a circle. The exact form of the equations, whileinteresting, may not be of as much importance as this qualitativeobservation which reflects the fact that the underlying phenomenon isrecurrent or periodic. When looking for periodic or recurrent phenomena,methods may be developed which can detect the presence of loops withoutdefining explicit models. For example, periodicity may be detectablewithout having to first develop a fully accurate model of the dynamics.

FIG. 1C is an example graph of data sets whereby the data does not breakup into disconnected groups, but instead has a structure in which thereare lines (or flares) emanating from a central group. In This case, thedata also suggests the presence of three distinct groups, but theconnectedness of the data does not reflect this. This particular datathat is the basis for the example graph in FIG. 1C arises from a studyof single nucleotide polymorphisms (SNPs).

In each of the examples above, aspects of the shape of the data arerelevant in reflecting information about the data. Connectedness (thesimplest property of shape) reflects the presence of a discreteclassification of the data into disparate groups. The presence of loops,another simple aspect of shape, often reflect periodic or recurrentbehavior. Finally, in the third example, the shape containing flaressuggests a classification of the data descriptive of ways in whichphenomena can deviate from the norm, which would typically berepresented by the central core. These examples support the idea thatthe shape of data (suitably defined) is an important aspect of itsstructure, and that it is therefore important to develop methods foranalyzing and understanding its shape. The part of mathematics whichconcerns itself with the study of shape is called topology, andtopological data analysis attempts to adapt methods for studying shapewhich have been developed in pure mathematics to the study of the shapeof data, suitably defined.

One question is how notions of geometry or shape are translated intoinformation about point clouds, which are, after all, finite sets? Whatwe mean by shape or geometry can come from a dissimilarity function ormetric (e.g., a non-negative, symmetric, real-valued function d on theset of pairs of points in the data set which may also satisfy thetriangle inequality, and d(x; y)=0 if and only if x=y). Such functionsexist in profusion for many data sets. For example, when data comes inthe form of a numerical matrix, where the rows correspond to the datapoints and the columns are the fields describing the data, then-dimensional Euclidean distance function is natural when there are nfields. Similarly, in this example, there are Pearson correlationdistances, cosine distances, and other choices.

When the data is not Euclidean, for example if one is consideringgenomic sequences, various notions of distance may be defined usingmeasures of similarity based on Basic Local Alignment Search Tool(BLAST) type similarity scores. Further, a measure of similarity cancome in non-numeric forms, such as social networks of friends orsimilarities of hobbies, buying patterns, tweeting, and/or professionalinterests. In any of these ways the notion of shape may be formulatedvia the establishment of a useful notion of similarity of data points.

One of the advantages of TDA is that TDA may depend on nothing more thansuch a notion, which is a very primitive or low-level model. TDA mayrely on many fewer assumptions than standard linear or algebraic models,for example. Further, the methodology may provide new ways ofvisualizing and compressing data sets, which facilitate understandingand monitoring data. The methodology may enable study ofinterrelationships among disparate data sets and/ormultiscale/multiresolution study of data sets. Moreover, the methodologymay enable interactivity in the analysis of data, using point and clickmethods.

In some embodiments, TDA may be a very useful complement to moretraditional methods, such as Principal Component Analysis (PCA),multidimensional scaling, and hierarchical clustering. These existingmethods are often quite useful, but suffer from significant limitations.PCA, for example, is an essentially linear procedure and there aretherefore limits to its utility in highly non-linear situations.Multidimensional scaling is a method which is not intrinsically linear,but can in many situations wash out detail, since it may overweightlarge distances. In addition, when metrics do not satisfy an intrinsicflatness condition, it may have difficulty in faithfully representingthe data. Hierarchical clustering does exhibit multiscale behavior, butrepresents data only as disjoint clusters, rather than retaining any ofthe geometry of the data set. In all four cases, these limitationsmatter for many varied kinds of data.

We now summarize example properties of an example construction, in someembodiments, which may be used for representing the shape of data setsin a useful, understandable fashion as a finite graph:

-   -   The input may be a collection of data points equipped in some        way with a distance or dissimilarity function, or other        description. This can be given implicitly when the data is in        the form of a matrix, or explicitly as a matrix of distances or        even the generating edges of a mathematical network.    -   One construction may also use one or more lens functions (i.e.        real valued functions on the data). Lens function(s) may depend        directly on the metric. For example, lens function(s) might be        the result of a density estimator or a measure of centrality or        data depth. Lens function(s) may, in some embodiments, depend on        a particular representation of the data, as when one uses the        first one or two coordinates of a principal component or        multidimensional scaling analysis. In some embodiments, the lens        function(s) may be columns which expert knowledge identifies as        being intrinsically interesting as in cholesterol levels and BMI        in a study of heart disease.    -   In some embodiments, the construction may depend on a choice of        two or more processing parameters, resolution, and gain.        Increase in resolution typically results in more nodes and an        increase in the gain increases the number of edges in a        visualization and/or graph in a reference space as further        described herein.    -   The output may be, for example, a visualization (e.g., a display        of connected nodes or “network”) or simplicial complex. One        specific combinatorial formulation in one embodiment may be that        the vertices form a finite set, and then the additional        structure may be a collection of edges (unordered pairs of        vertices) which are pictured as connections in this network.

In various embodiments, a system for handling, analyzing, andvisualizing data using drag and drop methods as opposed to text basedmethods is described herein. Philosophically, data analytic tools arenot necessarily regarded as “solvers,” but rather as tools forinteracting with data. For example, data analysis may consist of severaliterations of a process in which computational tools point to regions ofinterest in a data set. The data set may then be examined by people withdomain expertise concerning the data, and the data set may then besubjected to further computational analysis. In some embodiments,methods described herein provide for going back and forth betweenmathematical constructs, including interactive visualizations (e.g.,graphs), on the one hand and data on the other.

In one example of data analysis in some embodiments described herein, anexemplary clustering tool is discussed which may be more powerful thanexisting technology, in that one can find structure within clusters andstudy how clusters change over a period of time or over a change ofscale or resolution.

An example interactive visualization tool (e.g., a visualization modulewhich is further described herein) may produce combinatorial output inthe form of a graph which can be readily visualized. In someembodiments, the example interactive visualization tool maybe lesssensitive to changes in notions of distance than current methods, suchas multidimensional scaling.

Some embodiments described herein permit manipulation of the data from avisualization. For example, portions of the data which are deemed to beinteresting from the visualization can be selected and converted intodatabase objects, which can then be further analyzed. Some embodimentsdescribed herein permit the location of data points of interest withinthe visualization, so that the connection between a given visualizationand the information the visualization represents may be readilyunderstood.

FIG. 2 is an example environment 200 in which embodiments may bepracticed. In various embodiments, data analysis and interactivevisualization may be performed locally (e.g., with software and/orhardware on a local digital device), across a network (e.g., via cloudcomputing), or a combination of both. In many of these embodiments, adata structure is accessed to obtain the data for the analysis, theanalysis is performed based on properties and parameters selected by auser, and an interactive visualization is generated and displayed. Thereare many advantages between performing all or some activities locallyand many advantages of performing all or some activities over a network.

Environment 200 comprises user devices 202 a-202 n, a communicationnetwork 204, data storage server 206, and analysis server 208.Environment 200 depicts an embodiment wherein functions are performedacross a network. In this example, the user(s) may take advantage ofcloud computing by storing data in a data storage server 206 over acommunication network 204. The analysis server 208 may perform analysisand generation of an interactive visualization.

User devices 202 a-202 n may be any digital devices. A digital device isany device that includes memory and a processor. Digital devices arefurther described in FIG. 18. The user devices 202 a-202 n may be anykind of digital device that may be used to access, analyze and/or viewdata including, but not limited to a desktop computer, laptop, notebook,or other computing device.

In various embodiments, a user, such as a data analyst, may generateand/or receive a database or other data structure with the user device202 a to be saved to the data storage server 206. The user device 202 amay communicate with the analysis server 208 via the communicationnetwork 204 to perform analysis, examination, and visualization of datawithin the database.

The user device 202 a may comprise any number of client programs. One ormore of the client programs may interact with one or more applicationson the analysis server 208. In other embodiments, the user device 202 amay communicate with the analysis server 208 using a browser or otherstandard program. In various embodiments, the user device 202 acommunicates with the analysis server 208 via a virtual private network.Those skilled in the art will appreciate that that communication betweenthe user device 202 a, the data storage server 206, and/or the analysisserver 208 may be encrypted or otherwise secured.

The communication network 204 may be any network that allows digitaldevices to communicate. The communication network 204 may be theInternet and/or include LAN and WANs. The communication network 204 maysupport wireless and/or wired communication.

The data storage server 206 is a digital device that is configured tostore data. In various embodiments, the data storage server 206 storesdatabases and/or other data structures. The data storage server 206 maybe a single server or a combination of servers. In one example the datastorage server 206 may be a secure server wherein a user may store dataover a secured connection (e.g., via https). The data may be encryptedand backed-up. In some embodiments, the data storage server 206 isoperated by a third-party such as Amazon's S3 service.

The database or other data structure may comprise large high-dimensionaldatasets. These datasets are traditionally very difficult to analyzeand, as a result, relationships within the data may not be identifiableusing previous methods. Further, previous methods may be computationallyinefficient.

The analysis server 208 may include any number of digital devicesconfigured to analyze data (e.g., the data in the stored database and/orother dataset received and/or generated by the user device 202 a).Although only one digital device is depicted in FIG. 2 corresponding tothe analysis server 208, it will be appreciated that any number offunctions of the analysis server 208 may be performed by any number ofdigital devices.

In various embodiments, the analysis server 208 may perform manyfunctions to interpret, examine, analyze, and display data and/orrelationships within data. In some embodiments, the analysis server 208performs, at least in part, topological analysis of large datasetsapplying metrics, filters, and resolution parameters chosen by the user.The analysis is further discussed regarding FIG. 8 herein.

The analysis server 208 may generate graphs in memory, visualizedgraphs, and/or an interactive visualization of the output of theanalysis. The interactive visualization allows the user to observe andexplore relationships in the data. In various embodiments, theinteractive visualization allows the user to select nodes comprisingdata that has been clustered. The user may then access the underlyingdata, perform further analysis (e.g., statistical analysis) on theunderlying data, and manually reorient the graphs) (e.g., structures ofnodes and edges described herein) within the interactive visualization.The analysis server 208 may also allow for the user to interact with thedata, see the graphic result. The interactive visualization is furtherdiscussed in FIGS. 9-11.

The graphs in memory and/or visualized graphs may also include nodesand/or edges as described herein. Graphs that are generated in memorymay not be depicted to a user but rather may be in memory of a digitaldevice. Visualized graphs are rendered graphs that may be depicted tothe user (e.g., using user device 202 a).

In some embodiments, the analysis server 208 interacts with the userdevice(s) 202 a-202 n over a private and/or secure communicationnetwork. The user device 202 a may include a client program dial allowsthe user to interact with the data storage server 206, the analysisserver 208, another user device (e.g., user device 202 n), a database,and/or an analysis application executed on the analysis server 208.

It will be appreciated that all or part of the data analysis may occurat the user device 202 a. Further, all or part of the interaction withthe visualization (e.g., graphic) may be performed on the user device202 a. Alternately, all or part of the data analysis may occur on anynumber of digital devices including for example, on the analysis server208.

Although two user devices 202 a and 202 n are depicted, those skilled inthe art will appreciate that there may be any number of user devices inany location (e.g., remote from each other). Similarly, there may be anynumber of communication networks, data storage servers, and analysisservers.

Cloud computing may allow for greater access to large datasets (e.g.,via a commercial storage service) over a faster connection. Further,those skilled in the art will appreciate that services and computingresources offered to the user(s) may be scalable.

FIG. 3 is a block diagram of an example analysis server 208. In someembodiments, the analysis server 208 comprises a processor 302,input/output (I/O) interface 304, a communication network interface 306,a memory system 308, a storage system 310, and a processing module 312.The processor 302 may comprise any processor or combination ofprocessors with one or more cores.

The input/output (I/O) interface 304 may comprise interfaces for variousI/O devices such as, for example, a keyboard, mouse, and display device.The example communication network interface 306 is configured to allowthe analysis server 208 to communication with the communication network204 (see FIG. 2). The communication network interface 306 may supportcommunication over an Ethernet connection, a serial connection, aparallel connection, and/or an ATA connection. The communication networkinterface 306 may also support wireless communication (e.g., 802.11a/b/g/n, WiMax, LTE, WiFi). It will be apparent to those skilled in theart that the communication network interface 306 can support many wiredand wireless standards.

The memory system 308 may be any kind of memory including RAM, ROM, orflash, cache, virtual memory, etc. In various embodiments, working datais stored within the memory system 308. The data within the memorysystem 308 may be cleared or ultimately transferred to the storagesystem 310.

The storage system 310 includes any storage configured to retrieve andstore data. Some examples of the storage system 310 include flashdrives, hard drives, optical drives, and/or magnetic tape. Each of thememory system 308 and the storage system 310 comprises a non-transitorycomputer-readable medium, which stores instructions (e.g., softwareprograms) executable by processor 302.

The storage system 310 comprises a plurality of modules utilized byembodiments of discussed herein. A module may be hardware, software(e.g., including instructions executable by a processor), or acombination of both. In one embodiment, the storage system 310 includesa processing module 312. The processing module 312 may include memoryand/or hardware and includes an input module 314, a filter module 316, aresolution module 318, an analysis module 320, a visualization engine322, and database storage 324. Alternative embodiments of the analysisserver 208 and/or the storage system 310 may comprise more, less, orfunctionally equivalent components and modules.

The input module 314 may be configured to receive commands andpreferences from the user device 202 a. In various examples, the inputmodule 314 receives selections from the user which will be used toperform the analysis. The output of the analysis may be an interactivevisualization.

The input module 314 may provide the user a variety of interface windowsallowing the user to select and access a database, choose fieldsassociated with the database, choose a metric, choose one or morefilters, and identify resolution parameters for the analysis. In oneexample, the input module 314 receives a database identifier andaccesses a large multi-dimensional database. The input module 314 mayscan the database and provide the user with an interface window allowingthe user to identify an ID field. An ID field is an identifier for eachdata point. In one example, the identifier is unique. The same columnname may be present in the table from which filters are selected. Afterthe ID field is selected, the input module 314 may then provide the userwith another interface window to allow the user to choose one or moredata fields from a table of the database.

Although interactive windows may be described herein, those skilled inthe art will appreciate that any window, graphical user interface,and/or command line may be used to receive or prompt a user or userdevice 202 a for information.

The filter module 316 may subsequently provide the user with aninterface window to allow the user to select a metric to be used inanalysis of the data within the chosen data fields. The filter module316 may also allow the user to select and/or define one or more filters.

The resolution module 318 may allow the user to select a resolution,including filter parameters. In one example, the user enters a number ofintervals and a percentage overlap for a filter.

The analysis module 320 may perform data analysis based on the databaseand the information provided by the user. In various embodiments, theanalysis module 320 performs an algebraic topological analysis toidentify structures and relationships within data and clusters of data.Those skilled in the art will appreciate that the analysis module 320may use parallel algorithms or use generalizations of variousstatistical techniques (e.g., generalizing the bootstrap to zig-zagmethods) to increase the size of data sets that can be processed. Theanalysis is further discussed herein (e.g., see discussion regardingFIG. 8). It will be appreciated that the analysis module 320 is notlimited to algebraic topological analysis but may perform any analysis.

The visualization engine 322 generates an interactive visualizationbased on the output from the analysis module 320. The interactivevisualization allows the user to see all or part of the analysisgraphically. The interactive visualization also allows the user tointeract with the visualization. For example, the user may selectportions of a graph from within the visualization to see and/or interactwith the underlying data and/or underlying analysis. The user may thenchange the parameters of the analysis (e.g., change the metric,filters), or resolution(s)) which allows the user to visually identifyrelationships in the data that may be otherwise undetectable using priormeans. The interactive visualization is further described herein (e.g.,see discussion regarding FIGS. 9-11).

The database storage 324 is configured to store all or part of thedatabase that is being accessed. In some embodiments, the databasestorage 324 may store saved portions of the database. Further, thedatabase storage 324 may be used to store user preferences, parameters,and analysis output thereby allowing the user to perform many differentfunctions on the database without losing previous work.

Those skilled in the art will appreciate that that all or part of theprocessing module 312 may be at the user device 202 a or the databasestorage server 206. In some embodiments, all or some of thefunctionality of the processing module 312 may be performed by the userdevice 202 a.

In various embodiments, systems and methods discussed herein may beimplemented with one or more digital devices. In some examples, someembodiments discussed herein may be implemented by a computer program(instructions) executed by a processor. The computer program may providea graphical user interface. Although such a computer program isdiscussed, those skilled in the art will appreciate that embodiments maybe performed using any of the following, either alone or in combination,including, but not limited to, a computer program, multiple computerprograms, firmware, and/or hardware.

A module and/or engine may include any processor or combination ofprocessors. In some examples, a module and/or engine may include or be apart of a processor, digital signal processor (DSP), applicationspecific integrated circuit (ASIC), an integrated circuit, and/or thelike. In various embodiments, the module and/or engine may be softwareor firmware.

FIG. 4 is a flow chart 400 depicting an example method of datasetanalysis and visualization in some embodiments. In step 402, the inputmodule 314 accesses a database. The database may be any data structurecontaining data (e.g., a very large dataset of multidimensional data).In some embodiments, the database may be a relational database. In someexamples, the relational database may be used with MySQL, Oracle,Microsoft SQL Server, Aster nCluster, Teradata, and/or Vertica. Thoseskilled in the art will appreciate that the database may not be arelational database.

In some embodiments, the input module 314 receives a database identifierand a location of the database (e.g., the data storage server 206) fromthe user device 202 a (see FIG. 2). The input module 314 may then accessthe identified database. In various embodiments, the input module 314may read data from many different sources, including, but not limited toMS Excel files, text files (e.g., delimited or CSV), Matlab .mat format,or any other file.

In some embodiments, the input module 314 receives an IP address orhostname of a server hosting the database, a username, password, and thedatabase identifier. This information (herein referred to as “connectioninformation”) may be cached for later use. It will be appreciated thatthe database may be locally accessed and that all, some, or none of theconnection information may be required. In one example, the user device202 a may have full access to the database stored locally on the userdevice 202 a so the IP address is unnecessary. In another example, theuser device 202 a may already have loaded the database and the inputmodule 314 merely begins by accessing the loaded database.

In various embodiments, the identified database stores data withintables. A table may have a “column specification” which stores the namesof the columns and their data types. A “row” in a table, may be a tuplewith one entry for each column of the correct type. In one example, atable to store employee records might have a column specification suchas:

-   -   employee_id primary key int (this may store the employee's ID as        an integer, and uniquely identifies a row)    -   age int    -   gender char(1) (gender of the employee may be a single character        either M or F)    -   salary double (salary of an employee may be a floating point        number)    -   name varchar (name of the employee may be a variable-length        string)        In this example, each employee corresponds to a row in this        table. Further, the tables in this example relational database        are organized into logical units called databases. An analogy to        file systems is that databases can be thought of as folders and        files as tables. Access to databases may be controlled by the        database administrator by assigning a username/password pair to        authenticate users.

Once the database is accessed, the input module 314 may allow the userto access a previously stored analysis or to begin a new analysis. Ifthe user begins a new analysis, the input module 314 may provide theuser device 202 a with an interface window allowing the user to identifya table from within the database. In one example, the input module 314provides a list of available tables from the identified database.

In step 404, the input module 314 receives a table identifieridentifying a table from within the database. The input module 314 maythen provide the user with a list of available ID fields from the tableidentifier. In step 406, the input module 314 receives the ID fieldidentifier from the user and/or user device 202 a. The ID field is, insome embodiments, the primary key.

Having selected the primary key, the input module 314 may generate a newinterface window to allow the user to select data fields for analysis.In step 408, the input module 314 receives data field identifiers fromthe user device 202 a. The data within the data fields may be lateranalyzed by the analysis module 320.

In step 408, the filter module 316 selects one or more filters. In someembodiments, the filter module 316 and/or the input module 314 generatesan interface window allowing the user of the user device 202 a optionsfor a variety of different metrics and filter preferences. The interfacewindow may be a drop down menu identifying a variety of distance metricsto be used in the analysis.

In some embodiments, the user selects and/or provides filteridentifier(s) to the filter module 316. The role of the filters in theanalysis is also further described herein. The filters, for example, maybe user defined, geometric, or based on data which has beenpre-processed. In some embodiments, the data based filters are numericalarrays which can assign a set of real numbers to each row in the tableor each point in the data generally.

A variety of geometric filters may be available for the user to choose.Geometric filters may include, but are not limited to:

-   -   Density    -   L1 Eccentricity    -   L-infinity Eccentricity    -   Witness based Density    -   Witness based Eccentricity    -   Eccentricity as distance from a fixed point    -   Approximate Kurtosis of the Eccentricity

In step 410, the filter module 316 identifies a metric. Metric optionsmay include, but are not limited to, Euclidean, DB Metric, variancenormalized Euclidean, and total normalized Euclidean. The metric and theanalysis are further described herein.

In step 412, the resolution module 318 defines the resolution to be usedwith a filter in the analysis. The resolution may comprise a number ofintervals and an overlap parameter. In various embodiments, theresolution module 318 allows the user to adjust the number of intervalsand overlap parameter (e.g., percentage overlap) for one or morefilters.

In step 414, the analysis module 320 processes data of selected fieldsbased on the metric, filter(s), and resolution(s) to generate thevisualization. This process is further discussed herein (e.g., seediscussion regarding FIG. 8).

In step 416, the visualization engine 322 displays the interactivevisualization. In various embodiments, the visualization may be renderedin two or three dimensional space. The visualization engine 322 may usean optimization algorithm for an objective function which is correlatedwith good visualization (e.g., the energy of the embedding). Thevisualization may show a collection of nodes corresponding to each ofthe partial clusters in the analysis output and edges connecting them asspecified by the output The interactive visualization is furtherdiscussed herein (e.g., see discussion regarding FIGS. 9-11).

Although many examples discuss the input module 314 as providinginterface windows, it will be appreciated that all or some of theinterface may be provided by a client on the user device 202 a. Further,in some embodiments, the user device 202 a may be running all or some ofthe processing module 312.

FIGS. 5-7 depict various interface windows to allow the user to makeselections, enter information (e.g., fields, metrics, and filters),provide parameters (e.g., resolution), and provide data (e.g., identifythe database) to be used with analysis. It will be appreciated that anygraphical user interface or command line may be used to make selections,enter information, provide parameters, and provide data.

FIG. 5 is an exemplary ID field selection interface window 500 in someembodiments. The ID field selection interface window 500 allows the userto identify an ID field. The ID field selection interface window 500comprises a table search field 502, a table list 504, and a fieldsselection window 506.

In various embodiments, the input module 314 identifies and accesses adatabase from the database storage 324, user device 202 a, or the datastorage server 206. The input module 314 may then generate the ID fieldselection interface window 500 and provide a list of available tables ofthe selected database in the table list 504. The user may click on atable or search for a table by entering a search query (e.g., a keyword)in the table search field 502. Once a table is identified (e.g., clickedon by the user), the fields selection window 506 may provide a list ofavailable fields in the selected table. The user may then choose a fieldfrom the fields selection window 506 to be the ID field. In someembodiments, any number of fields may be chosen to be the ID field(s).

FIG. 6a is an example data field selection interface window 600 a insome embodiments. The data field selection interface window 600 a allowsthe user to identify data fields. The data field selection interfacewindow 600 a comprises a table search field 502, a table fist 504, afields selection window 602, and a selected window 604.

In various embodiments, after selection of the ID field, the inputmodule 314 provides a list of available tables of the selected databasein the table list 504. The user may click on a table or search for atable by entering a search query (e.g., a keyword) in the table searchfield 502. Once a table is identified (e.g., clicked on by the user),the fields selection window 506 may provide a list of available fieldsin the selected table. The user may then choose any number of fieldsfrom the fields selection window 602 to be data fields. The selecteddata fields may appear in the selected window 604. The user may alsodeselect fields that appear in the selected window 604.

Those skilled in the art will appreciate that the table selected by theuser in the table list 504 may be the same table selected with regard toFIG. 5. In some embodiments, however, the user may select a differenttable. Further, the user may, in various embodiments, select fields froma variety of different tables.

FIG. 6b is an example metric and filter selection interface window 600 bin some embodiments. The metric and filter selection interface window600 b allows the user to identify a metric, add filters), and adjustfilter parameters. The metric and filter selection interface window 600b comprises a metric pull down menu 606, an add filter from databaseburton 608, and an add geometric filter button 610.

In various embodiments, the user may click on the metric pull down menu606 to view a variety of metric options. Various metric options aredescribed herein. In some embodiments, the user may define a metric. Theuser defined metric may then be used with the analysis.

In one example, finite metric space data may be constructed from a datarepository (i.e., database, spreadsheet, or Matlab file). This may meanselecting a collection of fields whose entries will specify the metricusing the standard Euclidean metric for these fields, when they arefloating point or integer variables. Other notions of distance, such asgraph distance between collections of points, may be supported.

The analysis module 320 may perform analysis using the metric as a partof a distance function. The distance function can be expressed by aformula, a distance matrix, or other routine which computes it. The usermay add a filter from a database by clicking on the add filter fromdatabase button 608. The metric space may arise from a relationaldatabase, a Matlab file, an Excel spreadsheet, or other methods forstoring and manipulating data. The metric and filter selection interfacewindow 600 b may allow the user to browse for other filters to use inthe analysis. The analysis and metric function are further describedherein (e.g., see discussion regarding FIG. 8).

The user may also add a geometric filter 610 by clicking on the addgeometric filter button 610. In various embodiments, the metric andfilter selection interface window 600 b may provide a fist of geometricfilters from which the user may choose.

FIG. 7 is an example filter parameter interface window 700 in someembodiments. The filter parameter interface window 700 allows the userto determine a resolution for one or more selected filters (e.g.,filters selected in the metric and filter selection interface window600). The filter parameter interface window 700 comprises a filter namemenu 702, an interval field 704, an overlap bar 706, and a done button708.

The filter parameter interface window 700 allows the user to select afilter from the filter name menu 702. In some embodiments, the filtername menu 702 is a drop down box indicating all filters selected by theuser in the metric and filter selection interface window 600. Once afilter is chosen, the name of the filter may appear in the filter namemenu 702. The user may then change the intervals and overlap for one,some, or all selected filters.

The interval field 704 allows the user to define a number of intervalsfor the filter identified in the filter name menu 702. The user mayenter a number of intervals or scroll up or down to get to a desirednumber of intervals. Any number of intervals may be selected by theuser. The function of the intervals is further discussed herein (e.g.,see discussion regarding FIG. 8).

The overlap bar 706 allows the user to define the degree of overlap ofthe intervals for the filter identified in the filter name menu 702. Inone example, the overlap bar 706 includes a slider that allows the userto define the percentage overlap for the interval to be used with theidentified filter. Any percentage overlap may be set by the user.

Once the intervals and overlap are defined for the desired filters, theuser may click the done button. The user may then go back to the metricand filter selection interface window 600 and sec a new option to runthe analysis. In some embodiments, the option to run the analysis may beavailable in the filter parameter interface window 700. Once theanalysis is complete, the result may appear in an interactivevisualization further described herein (e.g., see discussion regardingFIGS. 9-11).

It will be appreciated that interface windows in FIGS. 4-7 are examples.The example interface windows are not limited to the functional objects(e.g., buttons, pull down menus, scroll fields, and search fields)shown. Any number of different functional objects may be used. Further,as described herein, any other interface, command line, or graphicaluser interface may be used.

FIG. 8 is a flowchart 800 for data analysis and generating aninteractive visualization in some embodiments. In various embodiments,the processing on data and user-specified options is motivated bytechniques from topology and, in some embodiments, algebraic topology.These techniques may be robust and general. In one example, thesetechniques apply to almost any kind of data for which some qualitativeidea of “closeness” or “similarity” exists. The techniques discussedherein may be robust because the results may be relatively insensitiveto noise in the data and even to errors in the specific details of thequalitative measure of similarity, which, in some embodiments, may begenerally refer to as “the distance function” or “metric.” It will beappreciated that while the description of the algorithms below may seemgeneral, the implementation of techniques described herein may apply toany level of generality.

In step 802, the input module 314 receives data S. In one example, auser identifies a data structure and then identifies ID and data fields.Data S may be based on the information within the ID and data fields. Invarious embodiments, data S is treated as being processed as a finite“similarity space,” where data S has a real-valued function d defined onpairs of points s and t in S, such that:d(s, s)=0d(s, t)=d(t, s)d(s, t)>=0These conditions may be similar to requirements for a finite metricspace, but the conditions may be weaker. In various examples, thefunction is a metric.

It will be appreciated that data S may be a finite metric space, or ageneralization thereof, such as a graph or weighted graph. In someembodiments, data S be specified by a formula, an algorithm, or by adistance matrix which specifies explicitly every pairwise distance.

In step 804, the input module 314 generates reference space R. In oneexample, reference space R may be a well-known metric space (e.g., suchas the real line). The reference space R may be defined by the user. Instep 806, the analysis module 320 generates a map ref( ) from S into R.The map ref( ) from S into R may be called the “reference map.”

In one example, a reference of map from S is to a reference metric spaceR. R may be Euclidean space of some dimension, but it may also be thecircle, torus, a tree, or other metric space. The map can be describedby one or more filters (i.e., real valued functions on S). These filterscan be defined by geometric invariants, such as the output of a densityestimator, a notion of data depth, or functions specified by the originof S as arising from a data set

In step 808, the resolution module 318 generates a cover of R based onthe resolution received from the user (e.g., filter(s), intervals, andoverlap—see discussion regarding FIG. 7 for example). The cover of R maybe a finite collection of open sets (in the metric of R) such that everypoint in R lies in at least one of these sets. In various examples, R isk-dimensional Euclidean space, where k is the number of filterfunctions. More precisely in this example. R is a box in k-dimensionalEuclidean space given by the product of the intervals [min_k, max_k] ,where min_k is the minimum value of the k-th filter function on S, andmax_k is the maximum value.

For example, suppose there are 2 filter functions, F1 and F2, and thatF1's values range from =1 to +1, and F2's values range from 0 to 5. Thenthe reference space is the rectangle in the x/y plane with corners(−1,0), (1,0), (−1, 5), (1, 5), as every point s of S will give rise toa pair (F1(s), F2(s)) that lies within that rectangle.

In various embodiments, the cover of R is given by taking products ofintervals of the covers of [min_k,max_k] for each of the k filters. Inone example, if the user requests 2 intervals and a 50% overlap for F1,the cover of the interval [−1,1] will be the two intervals (−1.5, 0.5),(−0.5, 1.5). If the user requests 5 intervals and a 30% overlap for F2,then that cover of [0, 5] will be (−3, 1.3), (0.7, 2.3), (1.7, 3.3),(2.7, 4.3), (3.7, 5.3). These intervals may give rise to a cover of the2-dimensional box by taking all possible pairs of intervals where thefirst of the pair is chosen from the cover for F1 and the second fromthe cover for F2. This may give rise to 2*5, or 10, open boxes thatcovered the 2-dimensional reference space. However, those skilled in theart will appreciate that the intervals may not be uniform, or that thecovers of a k-dimensional box may not be constructed by products ofintervals. In some embodiments, there are many other choices ofintervals. Further, in various embodiments, a wide range of coversand/or more general reference spaces may be used.

In one example, given a cover, C₁, . . . , C_(m), of R. the referencemap is used to assign a set of indices to each point in S, which are theindices of the C_(j) such that ref(s) belongs to C_(j). This functionmay be called ref_tags(.s). In a language such as Java, ref_tags wouldbe a method that returned an int∥. Since the C's cover R in thisexample, ref(s) must lie in at least one of them, but the elements ofthe cover usually overlap one another, which means that points that“land near the edges” may well reside in multiple cover sets. Inconsidering the two filter example, if F1(s) is −0.99, and F2(s) is0.001, then ref(s) is (−0.99, 0.001), and this lies in the cover element(−1.5, 0.5)×(−0.3, 1.3). Supposing that was labeled C₁, the referencemap may assign s to the set {1}. On the other hand, if t is mapped byF1, F2 to (0.1, 2.1), then ref(t) will be in (−1.5,0.5)×(0.7,2.3),(−0.5,1.5)×(0.7,2.3), (−1.5,0.5)×(1.7,3.3), and (−0.5,1.5)×(1.7,3.3), sothe set of indices would have four elements for t.

Having computed, for each point, which “cover tags” it is assigned to,for each cover element, C_(d), the points may be constructed, whose tagsincluded, as set S(d). This may mean that every point s is in S(d) forsome d, but some points may belong to more than one such set. In someembodiments, there is, however, no requirement that each S(d) isnon-empty, and it is frequently the case that some of these sets areempty. In the non-parallelized version of some embodiments, each point xis processed in turn, and x is inserted into a hash-bucket for each j inref_tags(t) (that is, this may be how S(d) sets are computed).

It will be appreciated that the cover of the reference space R may becontrolled by the number of intervals and the overlap identified in theresolution (e.g., see further discussion regarding FIG. 7). For example,the more intervals, the finer the resolution in S—that is, the fewerpoints in each S(d), but the more similar (with respect to the filters)these points may be. The greater the overlap, the more times thatclusters in S(d) may intersect clusters in S(e)—this means that more“relationships” between points may appear, but, in some embodiments, thegreater the overlap, the more likely that accidental relationships mayappear.

In step 810, the analysis module 320 clusters each S(d) based on themetric, filter, and the space S. In some embodiments, a dynamicsingle-linkage clustering algorithm may be used to partition S(d). Itwill be appreciated that any number of clustering algorithms may be usedwith embodiments discussed herein. For example, the clustering schememay be k-means clustering for some k, single linkage clustering, averagelinkage clustering, or any method specified by the user.

The significance of the user-specified inputs may now be seen. In someembodiments, a filter may amount to a “forced stretching” in a certaindirection. In some embodiments, the analysis module 320 may not clustertwo points unless ALL of the filter values are sufficiently “related”(recall that while normally related may mean “close,” the cover mayimpose a much more general relationship on the filter values, such asrelating two points s and t if ref(s) and ref(t) are sufficiently closeto the same circle in the plane). In various embodiments, the ability ofa user to impose one or more “critical measures” makes this techniquemore powerful than regular clustering, and the fact that these filterscan be anything, is what makes it so general.

The output may be a simplicial complex, from which one can extract its1-skeleton. The nodes of the complex may be partial clusters, (i.e.,clusters constructed from subsets of S specified as the preimages ofsets in the given covering of the reference space R).

In step 812, the visualization engine 322 identifies nodes which areassociated with a subset of the partition elements of all of the S(d)for generating an interactive visualization. For example, suppose thatS={1,2,3,4}, and the cover is C₁, C₂, C₃. Then if ref_tags(1)={1,2,3}and ref_tags(2)={2,3}, and ref_tags(3)={3}, and finally ref_tags(4)={1,3}, then S(1) in this example is {1, 4}, S(2)={1,2}, and S(3)={1,2,3,4}.If 1 and 2 are close enough to be clustered, and 3 and 4 are. butnothing else, then the clustering for S(1) may be {1} {3}, and for S(2)it may be {1,2}, and for S(3) it may be {1,2}, {3,4}. So the generatedgraph has, in this example, at most four nodes, given by the sets {1},{4}, {1,2}, and {3,4}(note that {1,2}appears in two differentclusterings). Of the sets of points that are used, two nodes intersectprovided that the associated node sets have a non-empty intersection(although this could easily be modified to allow users to require thatthe intersection is “large enough” either in absolute or relativeterms).

Nodes may be eliminated for any number of reasons. For example, a nodemay be eliminated as having too few points and/or not being connected toanything else. In some embodiments, the criteria for the elimination ofnodes (if any) may be under user control or have application-specificrequirements imposed on it. For example, if the points are consumers,for instance, clusters with too few people in area codes served by acompany could be eliminated. If a cluster was found with “enough”customers, however, this might indicate that expansion into area codesof the other consumers in the cluster could be warranted.

In step 814, the visualization engine 322 joins clusters to identifyedges (e.g., connecting lines between nodes). Once the nodes areconstructed, the intersections (e.g., edges) may be computed “all atonce,” by computing, for each point, the set of node sets (not ref_tags,this time). That is, for each s in S, node_id_set(s) may be computed,which is an int[]. In some embodiments, if the cover is well behaved,then this operation is linear in the size of the set S, and we theniterate over each pair in node_id_set(s). There may be an edge betweentwo node_id's if they both belong to the same node_id_set( ) value, andthe number of points in the intersection is precisely the number ofdifferent node_id sets in which that pair is seen. This means that,except for the clustering step (which is often quadratic in the size ofthe sets S(d), but whose size may be controlled by the choice of cover),all of the other steps in the graph construction algorithm may be linearin the size of S, and may be computed quite efficiently.

In step 816, the visualization engine 322 generates the interactivevisualization of interconnected nodes (e.g., nodes and edges displayedin FIGS. 9 and 10).

It will be appreciated that it is possible, in some embodiments, to makesense in a fairly deep way of connections between various ref( ) mapsand/or choices of clustering. Further, in addition to computing edges(pairs of nodes), the embodiments described herein may be extended tocompute triples of nodes, etc. For example, the analysis module 320 maycompute simplicial complexes of any dimension (by a variety of rules) onnodes, and apply techniques from homology theory to the graphs to helpusers understand a structure in an automatic (or semi-automatic) way.

Further, it will be appreciated that uniform intervals in the coveringmay not always be a good choice. For example, if the points areexponentially distributed with respect to a given filter, uniformintervals can fail—in such case adaptive interval sizing may yielduniformly-sized S(d) sets, for instance.

Further, in various embodiments, an interface may be used to encodetechniques for incorporating third-party extensions to data access anddisplay techniques. Further, an interface may be used to for third-partyextensions to underlying infrastructure to allow for new methods forgenerating coverings, and defining new reference spaces.

FIG. 9 is an example interactive visualization 900 in some embodiments.The display of the interactive visualization may be considered a “graph”in the mathematical sense. The interactive visualization comprises oftwo types of objects: nodes (e.g., nodes 902 and 906) (which may beballs and may be colored) and the edges (e.g., edge 904) (the blacklines). The edges connect pairs of nodes (e.g., edge 904 connects node902 with node 906). As discussed herein, each node may represent acollection of data points (rows in the database identified by the user).In one example, connected nodes tend to include data points which are“similar to”(e.g., clustered with) each other. The collection of datapoints may be referred to as being “in the node.” The interactivevisualization may be two-dimensional, three-dimensional, or acombination of both.

In various embodiments, connected nodes and edges may form a graph orstructure. There may be multiple graphs in the interactivevisualization. In one example, the interactive visualization may displaytwo or more unconnected structures of nodes and edges.

The visual properties of the nodes and edges (such as, but not limitedto, color, stroke color, text, texture, shape, coordinates of the nodeson the screen) can encode any data based property of the data pointswithin each node. For example, coloring of the nodes and/or the edgesmay indicate (but is not limited to) the following:

-   -   Values of fields or filters        -   Any general functions of the data in the nodes (e.g., if the            data were unemployment rates by state, then GDP of the            states may be identifiable by color the nodes)    -   Number of data points in the node

The interactive visualization 900 may contain a “bar” 910 which maycomprise a legend indicating patterns and/or coloring of the nodes(e.g., balls) and may also identify what the patterns and/or colorsindicate. For example, in FIG. 9, bar 910 may indicate that color ofsome nodes is based on the density filter with blue (on the far left ofthe bar 910) indicating “4.99e+03” and red (on the far right of the bar910) indicating “1.43e+04.” In general this might be expanded to showany other legend by which nodes and/or edges are colored. It will beappreciated that, in some embodiments, the user may control the color aswell as what the color (and/or stroke color, text, texture, shape,coordinates of the nodes on the screen) indicates.

The user may also drag and drop objects of the interactive visualization900. In various embodiments, the user may reorient structures of nodesand edges by dragging one or more nodes to another portion of theinteractive visualization (e.g., a window). In one example, the user mayselect node 902, hold node 902, and drag the node across the window. Thenode 902 will follow the user's cursor, dragging the structure of edgesand/or nodes either directly or indirectly connected to the node 902. Insome embodiments, the interactive visualization 900 may depict multipleunconnected structures. Each structure may include nodes, however, noneof the nodes of either structure are connected to each other. If theuser selects and drags a node of the first structure, only the firststructure will be reoriented with respect to the user action. The otherstructure will remain unchanged. The user may wish to reorient thestructure in order to view nodes, select nodes, and/or better understandthe relationships of the underlying data.

In one example, a user may drag a node to reorient the interactivevisualization (e.g., reorient the structure of nodes and edges). Whilethe user selects and/or drags the node, the nodes of the structureassociated with the selected node may move apart from each other inorder to provide greater visibility. Once the user lets go (e.g.,deselects or drops the node that was dragged), the nodes of thestructure may continue to move apart from each other.

In various embodiments, once the visualization engine 322 generates theinteractive display, the depicted structures may move by spreading outthe nodes from each other. In one example, the nodes spread from eachother slowly allowing the user to view nodes distinguish from each otheras well as the edges. In some embodiments, the visualization engine 322optimizes the spread of the nodes for the user's view. In one example,the structure(s) stop moving once an optimal view has been reached.

It will be appreciated that the interactive visualization 900 mayrespond to gestures (e.g., multi-touch), stylus, or other interactionsallowing the user to reorient nodes and edges and/or interacting withthe underlying data.

The interactive visualization 900 may also respond to user actions suchas when the user drags, clicks, or hovers a mouse cursor over a node. Insome embodiments, when the user selects a node or edge, node informationor edge information may be displayed. In one example, when a node isselected (e.g., clicked on by a user with a mouse or a mouse cursorhovers over the node), a node information box 908 may appear thatindicates information regarding the selected node. In this example, thenode information box 908 indicates an ID, box ID, number of elements(e.g., data points associated with the node), and density of the dataassociated with the node.

The user may also select multiple nodes and/or edges by clickingseparate on each object, or drawing a shape (such as a box) around thedesired objects. Once the objects are selected, a selection informationbox 912 may display some information regarding the selection. Forexample, selection information box 912 indicates the number of nodesselected and the total points (e.g., data points or elements) of theselected nodes.

The interactive visualization 900 may also allow a user to furtherinteract with the display. Color option 914 allows the user to displaydifferent information based on color of the objects. Color option 914 inFIG. 9 is set to filter_Density, however, other filters may be chosenand the objects re-colored based on the selection. It will beappreciated that the objects may be colored based on any filter,property of data, or characterization. When a new option is chosen inthe color option 914, the information and/or colors depicted in thecolor bar 910 may be updated to reflect the change.

Layout checkbox 916 may allow the user to anchor the interactivevisualization 900. In one example, the layout checkbox 916 is checkedindicating that the interactive visualization 900 is anchored. As aresult, the user will not be able to select and drag the node and/orrelated structure. Although other functions may still be available, thelayout checkbox 916 may help the user keep from accidentally movingand/or reorienting nodes, edges, and/or related structures. It will beappreciated the layout checkbox 916 may indicate that the interactivevisualization 900 is anchored when the layout checkbox 916 is uncheckedand that when the layout checkbox 916 is checked the interactivevisualization 900 is no longer anchored.

The change parameters button 918 may allow a user to change theparameters (e.g., add/remove filters and/or change the resolution of oneor more filters). In one example, when the change parameters button 918is activated, the user may be directed back to the metric and filterselection interface window 600 (see FIG. 6) which allows the user to addor remove filters (or change the metric). The user may then view thefilter parameter interface 700 (see FIG. 7) and change parameters (e.g.,intervals and overlap) for one or more filters. The analysis module 320may then re-analyze the data based on the changes and display a newinteractive visualization 900 without again having to specify the datasets, filters, etc.

The find ID's button 920 may allow a user to search for data within theinteractive visualization 900. In one example, the user may click thefind ID's button 920 and receive a window allowing the user to identifydata or identify a range of data. Data may be identified by ID orsearching for the data based on properties of data and/or metadata. Ifdata is found and selected, the interactive visualization 900 mayhighlight the nodes associated with the selected data. For example,selecting a single row or collection of rows of a database orspreadsheet may produce a highlighting of nodes whose correspondingpartial cluster contains any element of that selection.

In various embodiments, the user may select one or more objects andclick on the explain button 922 to receive in-depth informationregarding the selection. In some embodiments, when the user selects theexplain button 922, the information about the data from which theselection is based may be displayed. The function of the explain button922 is further discussed herein (e.g., see discussion regarding FIG.10).

In various embodiments, the interactive visualization 900 may allow theuser to specify and identify subsets of interest, such as outputfiltering, to remove clusters or connections which are too small orotherwise uninteresting. Further, the interactive visualization 900 mayprovide more general coloring and display techniques, including forexample, allowing a user to highlight nodes based on a user-specifiedpredicate, and coloring the nodes based on the intensity ofuser-specified weighting functions.

The interactive visualization 900 may comprise any number of menu items.The “Selection” menu may allow the following functions:

-   -   Select singletons (select nodes which are not connected to other        nodes)    -   Select all (selects all the nodes and edges)    -   Select all nodes (selects all nodes)    -   Select all edges    -   Clear selection (no selection)    -   Invert Selection (selects the complementary set of nodes or        edges)    -   Select “small” nodes (allows the user to threshold nodes based        on how many points they have)    -   Select leaves (selects all nodes which are connected to long        “chains” in the graph)    -   Remove selected nodes    -   Show in a table (shows the selected nodes and their associated        data in a table)    -   Save selected nodes (saves the selected data to whatever format        the user chooses. This may allow the user to subset the data and        create new data sources which may be used for further analysis.)

In one example of the “show in a table” option, information from aselection of nodes may be displayed. The information may be specific tothe origin of the data. In various embodiments, elements of a databasetable may be listed, however, other methods specified by the user mayalso be included. For example, in the case of microarray data from geneexpression data, heat maps may be used to view the results of theselections.

The interactive visualization 900 may comprise any number of menu items.The “Save” menu may allow may allow the user to save the whole output ina variety of different formats such as (but not limited to):

-   -   Image files (PNG/JPG/PDF/SVG etc.)    -   Binary output (The interactive output is saved in the binary        format The user may reopen this file at any time to get this        interactive window again)        In some embodiments, graphs may be saved in a format such that        the graphs may be used for presentations. This may include        simply saving the image as a pdf or png file, but it may also        mean saving an executable .xml file, which may permit other        users to use the search and save capability to the database on        the file without having to recreate the analysis.

In various embodiments, a relationship between a first and a secondanalysis output/interactive visualization for differing values of theinterval length and overlap percentage may be displayed. The formalrelationship between the first and second analysis output/interactivevisualization may be that when one cover refines the next, there is amap of simplicial complexes from the output of the first to the outputof the second. This can be displayed by applying a restricted form of athree-dimensional graph embedding algorithm, in which a graph is theunion of the graphs for the various parameter values and in which theconnections are the connections in the individual graphs as well asconnections from one node to its image in the following graph. Theconstituent graphs may be placed in its own plane in 3D space. In someembodiments, there is a restriction that each constituent graph remainwithin its associated plane. Each constituent graph may be displayedindividually, but a small change of parameter value may result in thevisualization of the adjacent constituent graph. In some embodiments,nodes in the initial graph will move to nodes in the next graph, in areadily visualizable way.

FIG. 10 is an example interactive visualization 1000 displaying anexplain information window 1002 in some embodiments. In variousembodiments, the user may select a plurality of nodes and click on theexplain button. When the explain button is clicked, the explaininformation window 1002 may be generated. The explain information window1002 may identify the data associated with the selected object(s) aswell as information (e.g., statistical information) associated with thedata.

In some embodiments, the explain button allows the user to get a sensefor which fields within the selected data fields are responsible for“similarity” of data in the selected nodes and the differentiatingcharacteristics. There can be many ways of scoring the data fields. Theexplain information window 1002 (i.e., the scoring window in FIG. 10) isshown along with the selected nodes. The highest scoring fields maydistinguish variables with respect to the rest of the data.

In one example, the explain information window 1002 indicates that datafrom fields day0-day6 has been selected. The minimum value of the datain all of the fields is 0. The explain information window 1002 alsoindicates the maximum values. For example, the maximum value of all ofthe data associated with the day0 field across all of the points of theselected nodes is 0.353. The average (i.e., mean) of all of the dataassociated with the day0 field across all of the points of the selectednodes is 0.031. The score may be a relative (e.g., normalized) valueindicating the relative function of the filter, here, the score mayindicate the relative density of the data associated with the day0 fieldacross all of the points of the selected nodes. Those skilled in the artwill appreciate that any information regarding the data and/or selectednodes may appear in the explain information window 1002.

It will be appreciated that the data and the interactive visualization1000 may be interacted with in any number of ways. The user may interactwith the data directly to see where the graph corresponds to the data,make changes to the analysis and view the changes in the graph, modifythe graph and view changes to the data, or perform any kind ofinteraction.

FIG. 11 is a flowchart 1100 of functionality of the interactivevisualization in some embodiments. In step 1102, the visualizationengine 322 receives the analysis from the analysis module 320 and graphsnodes as balls and edges as connectors between balls 1202 to createinteractive visualization 900 (see FIG. 9).

In step 1104, the visualization engine 322 determines if the user ishovering a mouse cursor over (or has selected) a ball (i.e., a node). Ifthe user is hovering a mouse cursor over a ball or is selecting a ball,then information may be displayed regarding the data associated with theball. In one example, the visualization engine 322 displays a nodeinformation window 908.

If the visualization engine 322 does not determine that the user ishovering a mouse cursor over (or has selected) a ball, then thevisualization engine 322 determines if the user has selected balls onthe graph (e.g., by clicking on a plurality of balls or drawing a boxaround a plurality of balls). If the user has selected a plurality ofballs on the graph, the visualization engine 322 may highlight theselected balls on the graph in step 1110. The visualization engine 322may also display information regarding the selection (e.g., bydisplaying a selection information window 912). The user may also clickon the explain button 922 to receive more information associated withthe selection (e.g., the visualization engine 322 may display theexplain information window 1002).

In step 1112, the user may save the selection. For example, thevisualization engine 322 may save the underlying data, selected metric,filters, and/or resolution. The user may then access the savedinformation and create a new structure in another interactivevisualization 900 thereby allowing the user to focus attention on asubset of the data.

If the visualization engine 322 does not determine that the user hasselected balls on the graph, the visualization engine 322 may determineif the user selects and drags a ball on the graph in step 1114. If theuser selects and drags a ball on the graph, the visualization engine 322may reorient the selected balls and any connected edges and balls basedon the user's action in step 1116. The user may reorient all or part ofthe structure at any level of granularity.

It will be appreciated that although FIG. 11 discussed the user hoveringover, selecting, and/or dragging a ball, the user may interact with anyobject in the interactive visualization 900 (e.g., the user may hoverover, select, and/or drag an edge). The user may also zoom in or zoomout using the interactive visualization 900 to focus on all or a part ofthe structure (e.g., one or more balls and/or edges). Any number ofactions and operations may be performed using the interactivevisualization 900.

Further, although balls are discussed and depicted in FIGS. 9-11, itwill be appreciated that the nodes may be any shape and appear as anykind of object. Further, although some embodiments described hereindiscuss an interactive visualization being generated based on the outputof algebraic topology, the interactive visualization may be generatedbased on any kind of analysis and is not limited.

For years, researchers have been collecting huge amounts of data onbreast cancer, yet we are still battling the disease. Complexity, ratherthan quantity, is one of the fundamental issues in extracting knowledgefrom data. A topological data exploration and visualization platform mayassist the analysis and assessment of complex data. In variousembodiments, a predictive and visual cancer map generated by thetopological data exploration and visualization platform may assistphysicians to determine treatment options.

In one example, a breast cancer map visualization may be generated basedon the large amount of available information already generated by manyresearchers. Physicians may send biopsy data directly to a cloud-basedserver which may localize a new patient's data within the breast cancermap visualization. The breast cancer map visualization may be annotated(e.g., labeled) such that the physician may view outcomes of patientswith similar profiles as well as different kinds of statisticalinformation such as survival probabilities. Each new data point from apatient may be incorporated into the breast cancer map visualization toimprove accuracy of the breast cancer map visualization over time.

Although the following examples are largely focused on cancer mapvisualizations, it will be appreciated that at least some of theembodiments described herein may apply to any biological condition andnot be limited to cancer and/or disease. For example, some embodiments,may apply to different industries.

FIG. 12 is a flowchart for generating a cancer map visualizationutilizing biological data of a plurality of patients in someembodiments. In various embodiments, the processing of data anduser-specified options is motivated by techniques from topology and, insome embodiments, algebraic topology. As discussed herein, thesetechniques may be robust and general. In one example, these techniquesapply to almost any kind of data for which some qualitative idea of“closeness” or “similarity” exists. It will be appreciated that theimplementation of techniques described herein may apply to any level ofgenerality.

In various embodiments, a cancer map visualization is generated usinggenomic data linked to clinical outcomes (i.e., medical characteristics)which may be used by physicians during diagnosis and/or treatment.Initially, publicly available data sets may be integrated to constructthe topological map visualizations of patients (e.g., breast cancerpatients). It will be appreciated that any private, public, orcombination of private and public data sets may be integrated toconstruct the topological map visualizations. A map visualization may bebased on biological data such as, but not limited to, gene expression,sequencing, and copy number variation. As such, the map visualizationmay comprise many patients with many different types of collected data.Unlike traditional methods of analysis where distinct studies of breastcancer appear as separate entities, the map visualization may fusedisparate data sets while utilizing many datasets and data types.

In various embodiments, a new patient may be localized on the mapvisualization. With the map visualization for subtypes of a particulardisease and a new patient diagnosed with the disease, point(s) may belocated among the data points used in computing the map visualization(e.g., nearest neighbor) which is closest to the new patient point. Thenew patient may be labeled with nodes in the map visualizationcontaining the closest neighbor. These nodes may be highlighted to givea physician the location of the new patient among the patients in thereference data set. The highlighted nodes may also give the physicianthe location of the new patient relative to annotated disease subtypes.

The visualization map may be interactive and/or searchable in real-timethereby potentially enabling extended analysis and providing speedyinsight into treatment.

In step 1202, biological data and clinical outcomes of previous patientsmay be received. The clinical outcomes may be medical characteristics.Biological data is any data that may represent a condition (e.g., amedical condition) of a person. Biological data may include any healthrelated, medical, physical, physiological, pharmaceutical dataassociated with one or more patients. In one example, biological datamay include measurements of gene expressions for any number of genes. Inanother example, biological data may include sequencing information(e.g., RNA sequencing).

In various embodiments, biological data for a plurality of patients maybe publicly available. For example, various medical health facilitiesand/or public entities may provide gene expression data for a variety ofpatients. In addition to the biological data, information regarding anynumber of clinical outcomes, treatments, therapies, diagnoses and/orprognoses may also be provided. Those skilled in the art will appreciatethat any kind of information may be provided in addition to thebiological data.

The biological data, in one example, may be similar to data S asdiscussed with regard to step 802 of FIG. 8. The biological data mayinclude ID fields that identify patients and data fields that arerelated to the biological information (e.g., gene expressionmeasurements).

FIG. 13 is an example data structure 1300 including biological data 1304a-1304 y for a number of patients 1308 a-1308 n that may be used togenerate the cancer map visualization in some embodiments. Column 1302represents different patient identifiers for different patients. Thepatient identifiers may be any identifier.

At least some biological data may be contained within gene expressionmeasurements 1304 a-1304 y. In FIG. 13, “y” represents any number. Forexample, there may be 50,000 or more separate columns for different geneexpressions related to a single patient or related to one or moresamples from a patient. It will be appreciated that column 1304 a mayrepresent a gene expression measurement for each patient (if any forsome patients) associated with the patient identifiers in column 1302.The column 1304 b may represent a gene expression measurement of one ormore genes that are different than that of column 1304 a. As discussed,there may be any number of columns representing different geneexpression measurements.

Column 1306 may include any number of clinical outcomes, prognoses,diagnoses, reactions, treatments, and/or any other informationassociated with each patient. All or some of the information containedin column 1306 may be displayed (e.g., by a label or an annotation thatis displayed on the visualization or available to the user of thevisualization via clicking) on or for the visualization.

Rows 1308 a-1308 n each contains biological data associated with thepatient identifier of the row. For example, gene expressions in row 1308a are associated with patient identifier P1. As similarly discussed withregard to “y” herein, “n” represents any number. For example, there maybe 100,000 or more separate rows for different patients.

It will be appreciated that there may be any number of data structuresthat contain any amount of biological data for any number of patients.The data structured) may be utilized to generate any number of mapvisualizations.

In step 1204, the analysis server may receive a filter selection. Insome embodiments, the filter selection is a density estimation function.It will be appreciated that the filter selection may include a selectionof one or more functions to generate a reference space.

In step 1206, the analysis server performs the selected filter(s) on thebiological data of the previous patients to map the biological data intoa reference space. In one example, a density estimation function, whichis well known in the art, may be performed on the biological data (e.g.,data associated with gene expression measurement data 1304 a-1304 y) torelate each patient identifier to one or more locations in the referencespace (e.g., on a real line).

In step 1208, the analysis server may receive a resolution selection.The resolution may be utilized to identify overlapping portions of thereference space (e.g., a cover of the reference space R) in step 1210.

As discussed herein, the cover of R may be a finite collection of opensets (in the metric of R) such that every point in R lies in at leastone of these sets. In various examples, R is k-dimensional Euclideanspace, where k is the number of filter functions. Those skilled in theart will appreciate that the cover of the reference space R may becontrolled by the number of intervals and the overlap identified in theresolution (e.g., see FIG. 7). For example, the more intervals, thefiner the resolution in S (e.g., the similarity space of the receivedbiological data) —that is, the fewer points in each S(d), but the moresimilar (with respect to the filters) these points may be. The greaterthe overlap, the more times that clusters in S(d) may intersect clustersin S(e)—this means that more “relationships” between points may appear,but, in some embodiments, the greater the overlap, the more likely thataccidental relationships may appear.

In step 1212, the analysis server receives a metric to cluster theinformation of the cover in the reference space to partition S(d). Inone example, the metric may be a Pearson Correlation. The clusters mayform the groupings (e.g., nodes or balls). Various cluster means may beused including, but not limited to, a single linkage, average linkage,complete linkage, or k-means method.

As discussed herein, in some embodiments, the analysis module 320 maynot cluster two points unless filter values are sufficiently “related”(recall that while normally related may mean “close,” the cover mayimpose a much more general relationship on the filter values, such asrelating two points s and t if ref(s) and ref(t) are sufficiently closeto the same circle in the plane where ref( ) represents one or morefilter functions). The output may be a simplicial complex, from whichone can extract its 1-skeleton. The nodes of the complex may be partialclusters, (i.e., clusters constructed from subsets of S specified as thepreimages of sets in the given covering of the reference space R).

In step 1214, the analysis server may generate the visualization mapwith nodes representing clusters of patient members and edges betweennodes representing common patient members. In one example, the analysisserver identifies nodes which are associated with a subset of thepartition elements of all of the S(d) for generating an interactivevisualization.

As discussed herein, for example, suppose that S={1,2,3,4}, and thecover is C₁, C₂, C₃. Suppose cover C₁ contains {1,4}, C₂ contains {1,2},and C₃ contains {1,2,3,4}. If 1 and 2 are close enough to be clustered,and 3 and 4 are, but nothing else, then the clustering for S(1) may be{1}, {4}, and for S(2) it may be {1,2}, and for S(3) it may be {1,2},{3,4}. So the generated graph has, in this example, at most four nodes,given by the sets {1}, {4}, {1,2}, and {3,4} (note that {1,2} appears intwo different clusterings). Of the sets of points data are used, twonodes intersect provided that the associated node sets have a non-emptyintersection (although this could easily be modified to allow users torequire that the intersection is “large enough” either in absolute orrelative terms).

As a result of clustering, member patients of a grouping may sharebiological similarities (e.g., similarities based on the biologicaldata).

The analysis server may join clusters to identify edges (e.g.,connecting lines between nodes). Clusters joined by edges (i.e.,interconnections) share one or more member patients. In step 1216, adisplay may display a visualization map with attributes based on theclinical outcomes contained in the data structures (e.g., see FIG. 13regarding clinical outcomes). Any labels or annotations may be utilizedbased on information contained in the data structures. For example,treatments, prognoses, therapies, diagnoses, and the like may be used tolabel the visualization. In some embodiments, the physician or otheruser of the map visualization accesses the annotations or labels byinteracting with the map visualization.

The resulting cancer map visualization may reveal interactions andrelationships that were obscured, untested, and/or previously notrecognized.

FIG. 14 is an example visualization displaying the cancer mapvisualization 1400 in some embodiments. The cancer map visualization1400 represents a topological network of cancer patients. The cancer mapvisualization 1400 may be based on publicly and/or privately availabledata.

In various embodiments, the cancer map visualization 1400 is createdusing gene expression profiles of excised tumors. Each node (i.e., ballor grouping displayed in the map visualization 1400) contains a subsetof patients with similar genetic profiles.

As discussed herein, one or more patients (i.e., patient members of eachnode or grouping) may occur in multiple nodes. A patient may share asimilar genetic profile with multiple nodes or multiple groupings. Inone example, of 50,000 different gene expressions of the biologicaldata, multiple patients may share a different genetic profiles (e.g.,based on different gene expression combinations) with differentgroupings. When a patient shares a similar genetic profile withdifferent groupings or nodes, the patient may be included within thegroupings or nodes.

The cancer map visualization 1400 comprises groupings andinterconnections that are associated with different clinical outcomes.All or some of the clinical outcomes may be associated with thebiological data that generated the cancer map visualization 1400. Thecancer map visualization 1400 includes groupings associated withsurvivors 1402 and groupings associated with non-survivors 1404. Thecancer map visualization 1400 also includes different groupingsassociated with estrogen receptor positive non-survivors 1406, estrogenreceptor negative non-survivors 1408, estrogen receptor positivesurvivors 1410, and estrogen receptor negative survivors 1412.

In various embodiments, when one or more patients are members of two ormore different nodes, the nodes are interconnected by an edge (e.g., aline or interconnection). If there is not an edge between the two nodes,then there are no common member patients between the two nodes. Forexample, grouping 1414 shares at least one common member patient withgrouping 1418. The intersection of the two groupings is represented byedge 1416. As discussed herein, the number of shared member patients ofthe two groupings may be represented in any number of ways includingcolor of the interconnection, color of the groupings, size of theinterconnection, size of the groupings, animations of theinterconnection, animations of the groupings, brightness, or the like.In some embodiments, the number and/or identifiers of shared memberpatients of the two groupings may be available if the user interactswith the groupings 1414 and/or 1418 (e.g., draws a box around the twogroupings and the interconnection utilizing an input device such as amouse).

In various embodiments, a physician, on obtaining some data on a breasttumor, direct the data to an analysis server (e.g., analysis server 208over a network such as the Internet) which may localize the patientrelative to one or more groupings on the cancer map visualization 1400.The context of the cancer map visualization 1400 may enable thephysician to assess various possible outcomes (e.g., proximity ofrepresentation of new patient to the different associations of clinicaloutcomes).

FIG. 15 is a flowchart of for positioning new patient data relative to acancer map visualization in some embodiments. In step 1502, newbiological data of a new patient is received. In various embodiments, aninput module 314 of an analysis server (e.g., analysis server 208 ofFIGS. 1 and 2) may receive biological data of a new patient from aphysician or medical facility that performed analysis of one or moresamples to generate the biological data. The biological data may be anydata that represents a biological data of the new patient including, forexample, gene expressions, sequencing information, or the like.

In some embodiments, the analysis server 208 may comprise a new patientdistance module and a location engine. In step 1504, the new patientdistance module determines distances between the biological data of eachpatient of the cancer map visualization 1600 and the new biological datafrom the new patient. For example, the previous biological data that wasutilized in the generation of the cancer map visualization 1600 may bestored in mapped data structures. Distances may be determined betweenthe new biological data of the new patient and each of the previouspatient's biological data in the mapped data structure.

It will be appreciated that distances may be determined in any number ofways using any number of different metrics or Junctions. Distances maybe determined between the biological data of the previous patients andthe new patients. For example, a distance may be determined between afirst gene expression measurement of the new patient and each (or asubset) of the first gene expression measurements of the previouspatients (e.g., the distance between G1 of the new patient and G1 ofeach previous patient may be calculated). Distances may be determinedbetween all (or a subset of) other gene expression measurements of thenew patient to the gene expression measurements of the previouspatients.

In various embodiments, a location of the new patient on the cancer mapvisualization 1600 may be determined relative to the other memberpatients utilizing the determined distances.

In step 1506, the new patient distance module may compare distancesbetween the patient members of each grouping to the distances determinedfor the new patient. The new patient may be located in the grouping ofpatient members that are closest in distance to the new patient In someembodiments, the new patient location may be determined to be within agrouping that contains the one or more patient members that are closestto the new patient (even if other members of the grouping have longerdistances with the new patient). In some embodiments, this step isoptional.

In various embodiments, a representative patient member may bedetermined for each grouping. For example, some or all of the patientmembers of a grouping may be averaged or otherwise combined to generatea representative patient member of the grouping (e.g., the distancesand/or biological data of the patient members may be averaged oraggregated). Distances may be determined between the new patientbiological data and the averaged or combined biological data of one ormore representative patient members of one or more groupings. Thelocation engine may determine the location of the new patient based onthe distances. In some embodiments, once the closest distance betweenthe new patient and the representative patient member is found,distances may be determined between the new patient and the individualpatient members of the grouping associated with the closestrepresentative patient member.

In optional step 1508, a diameter of the grouping with the one or moreof the patient members that are closest to the new patient (based on thedetermined distances) may be determined. In one example, the diametersof the groupings of patient members closest to the new patient arecalculated. The diameter of the grouping may be a distance between twopatient members who are the farthest from each other when compared tothe distances between all patient members of the grouping. If thedistance between the new patient and the closest patient member of thegrouping is less than the diameter of the grouping, the new patient maybe located within the grouping If the distance between the new patientand the closest patient member of the grouping is greater than thediameter of the grouping, the new patient may be outside the grouping(e.g., a new grouping may be displayed on the cancer map visualizationwith the new patient as the single patient member of the grouping). Ifthe distance between the new patient and the closest patient member ofthe grouping is equal to the diameter of the grouping, the new patientmay be placed within or outside the grouping.

It will be appreciated that the determination of the diameter of thegrouping is not required in determining whether the new patient locationis within or outside of a grouping. In various embodiments, adistribution of distances between member patients and between memberpatients and the new patient is determined. The decision to locate thenew patient within or outside of the grouping may be based on thedistribution. For example, if there is a gap in the distribution ofdistances, the new patient may be separated from the grouping (e.g., asa new grouping). In some embodiments, if the gap is greater than apreexisting threshold (e.g., established by the physician, other user,or previously programmed), the new patient may be placed in a newgrouping that is placed relative to the grouping of the closest memberpatients. The process of calculating the distribution of distances ofcandidate member patients to determine whether there may be two or moregroupings may be utilized in generation of the cancer map visualizationfurther described herein (e.g., in the process as described with regardto FIG. 12). It will be appreciated that there may be any number of waysto determine whether a new patient should be included within a groupingof other patient members.

In step 1510, the location engine determines the location of the newpatient relative to the member patients and/or groupings of the cancermap visualization. The new location may be relative to the determineddistances between the new patient and the previous patients. Thelocation of the new patient may be part of a previously existinggrouping or may form a new grouping.

In some embodiments, the location of the new patient with regard to thecancer map visualization may be performed locally to the physician. Forexample, the cancer map visualization 1400 may be provided to thephysician (e.g., via a digital device). The physician may load the newpatient's biological data locally and the distances may be determinedlocally or via a cloud-based server. The location(s) associated with thenew patient may be overlaid on the previously existing cancer mapvisualization either locally or remotely.

It will be appreciated that, in some embodiments, the previous state ofthe cancer map visualization (e.g., cancer map visualization 1400) maybe retained or otherwise stored and a new cancer map visualizationgenerated utilizing the new patient biological data (e.g., in a methodsimilar to that discussed with regard to FIG. 12). The newly generatedmap may be compared to the previous state and the differences may behighlighted thereby, in some embodiments, highlighting the location(s)associated with the new patient. In this way, distances may be not becalculated as described with regard to FIG. 15, but rather, the processmay be similar to that as previously discussed.

FIG. 16 is an example visualization displaying the cancer map includingpositions for three new cancer patients in some embodiments. The cancermap visualization 1400 comprises groupings and interconnections that areassociated with different clinical outcomes as discussed with regard toFIG. 14. All or some of the clinical outcomes may be associated with thebiological data that generated the cancer map visualization 1400. Thecancer map visualization 1400 includes different groupings associatedwith survivors 1402, groupings associated with non-survivors 1404,estrogen receptor positive non-survivors 1406, estrogen receptornegative non-survivors 1408, estrogen receptor positive survivors 1410,and estrogen receptor negative survivors 1412.

The cancer map visualization 1400 includes three locations for three newbreast cancer patients. The breast cancer patient location 1602 isassociated with the clinical outcome of estrogen receptor positivesurvivors. The breast cancer patient location 1604 is associated withthe clinical outcome of estrogen receptor negative survivors.Unfortunately, breast cancer patient location 1606 is associated withestrogen receptor negative non-survivors. Based on the locations, aphysician may consider different diagnoses, prognoses, treatments, andtherapies to maintain or attempt to move the breast cancer patient to adifferent location utilizing the cancer map visualization 1400.

In some embodiments, the physician may assess the underlying biologicaldata associated with any number of member patients of any number ofgroupings to better understand the genetic similarities and/ordissimilarities. The physician may utilize the information to makebetter informed decisions.

The patient location 1604 is highlighted on the cancer map visualization1400 as active (e.g., selected by the physician). It will be appreciatedthat the different locations may be of any color, size, brightness,and/or animated to highlight the desired location(s) for the physician.Further, although only one location is identified for three differentbreast cancer patients, any of the breast cancer patients may havemultiple locations indicating different genetic similarities.

It will be appreciated that the cancer map visualization 1400 may beupdated with new information at any time. As such, as new patients areadded to the cancer map visualization 1400, the new data updates thevisualization such that as future patients are placed in the map, themap may already include the updated information. As new informationand/or new patient data is added to the cancer map visualization 1400,the cancer map visualization 1400 may improve as a tool to better informphysicians or other medical professionals.

In various embodiments, the cancer map visualization 1400 may trackchanges in patients over time. For example, updates to a new patient maybe visually tracked as changes in are measured in the new patient'sbiological data. In some embodiments, previous patient data is similarlytracked which may be used to determine similarities of changes based oncondition, treatment, and/or therapies, for example. In variousembodiments, velocity of change and/or acceleration of change of anynumber of patients may be tracked over time using or as depicted on thecancer map visualization 1400. Such depictions may assist the treatingphysician or other personnel related to the treating physician to betterunderstand changes in the patient and provide improved, current, and/orupdated diagnoses, prognoses, treatments, and/or therapies.

FIG. 17 is a flowchart of utilization the visualization and positioningof new patient data in some embodiments. In various embodiments, aphysician may collect amounts of genomic information from tumors removedfrom a new patient, input the data (e.g., upload the data to an analysisserver), and receive a map visualization with a location of the newpatient. The new patient's location within the map may offer thephysician new information about the similarities to other patients. Insome embodiments, the map visualization may be annotated so that thephysician may check the outcomes of previous patients in a given regionof the map visualization are distributed and then use the information toassist in decision-making for diagnosis, treatment, prognosis, and/ortherapy.

In step 1702, a medical professional or other personnel may remove asample from a patient. The sample may be of a tumor, blood, or any otherbiological material. In one example, a medical professional performs atumor excision. Any number of samples may be taken from a patient.

In step 1704, the sample(s) may be provided to a medical facility todetermine new patient biological data. In one example, the medicalfacility measures genomic data such as gene expression of a number ofgenes or protein levels.

In step 1706, the medical professional or other entity associated withthe medical professional may receive the new patient biological databased on the sample(s) from the new patient. In one example, a physicianmay receive the new patient biological data. The physician may provideall or some of the new patient biological data to an analysis serverover the Internet (e.g., the analysis server may be a cloud-basedserver). In some embodiments, the analysis server is the analysis server208 of FIG. 2. In some embodiments, the medical facility that determinesthe new patient biological data provides the biological data in anelectronic format which may be uploaded to the analysis server. In someembodiments, the medical facility that determines the new patientbiological data (e.g., the medical facility that measures the genomicdata) provide the biological data to the analysis server at the requestof the physician or others associated with the physician. It will beappreciated that the biological data may be provided to the analysisserver in any number of ways.

The analysis server may be any digital device and may not be limited toa digital device on a network. In some embodiments, the physician mayhave access to the digital device. For example, the analysis server maybe a table, personal computer, local server, or any other digitaldevice.

Once the analysis server receives the biological data of the new patient(e.g., the new patient biological data may be uploaded to the analysisserer in step 1708), the new patient may be localized in the mapvisualization and the information may be sent back to the physician instep 1710. The visualization may be a map with nodes representingclusters of previous patient members and edges between nodesrepresenting common patient members. The visualization may furtherdepict one or more locations related to the biological data of the newpatient.

The map visualization may be provided to the physician or otherassociated with the physician in real-time. For example, once thebiological data associated with the new patient is provided to theanalysis server, the analysis server may provide the map visualizationback to the physician or other associated with the physician within areasonably short time (e.g., within seconds or minutes). In someembodiments, the physician may receive the map visualization over anytime.

The map visualization may be provided to the physician in any number ofways. For example, the physician may receive the map visualization overany digital device such as, but not limited to, an office computer,IPad, tablet device, media device, smartphone, e-reader, or laptop.

In step 1712, the physician may assess possible different clinicaloutcomes based on the map visualization. In one example, the map-aidedphysician may make decisions on therapy and treatments depending onwhere the patient lands on the visualization (e.g., survivor ornon-survivor). The map visualization may include annotations or labelsthat identify one or more sets of groupings and interconnections asbeing associated with one or more clinical outcomes. The physician mayassess possible clinical outcomes based on the position(s) on the mapassociated with the new patient.

FIG. 18 is a block diagram of an exemplary digital device 1800. Thedigital device 1800 comprises a processor 1802, a memory system 1804, astorage system 1806, a communication network interface 1808, an I/Ointerface 1810, and a display interface 1812 communicatively coupled toa bus 1814. The processor 1802 may be configured to execute executableinstructions (e.g., programs). In some embodiments, the processor 1802comprises circuitry or any processor capable of processing theexecutable instructions.

The memory system 1804 is any memory configured to store data. Someexamples of the memory system 1804 are storage devices, such as RAM orROM. The memory system 1804 can comprise the ram cache. In variousembodiments, data is stored within the memory system 1804. The datawithin the memory system 1804 may be cleared or ultimately transferredto the storage system 1806.

The storage system 1806 is any storage configured to retrieve and storedata. Some examples of the storage system 1806 are flash drives, harddrives, optical drives, and/or magnetic tape. In some embodiments, thedigital device 1800 includes a memory system 1804 in the form of RAM anda storage system 1806 in the form of flash data. Both the memory system1804 and the storage system 1806 comprise computer readable media whichmay store instructions or programs that are executable by a computerprocessor including the processor 1802.

The communication network interface (com. network interface) 1808 can becoupled to a data network (e.g., communication network 204) via the link1816. The communication network interface 1808 may support communicationover an Ethernet connection, a serial connection, a parallel connection,or an ATA connection, for example. The communication network interface1808 may also support wireless communication (e.g., 1802.11 a/b/g/n,WiMAX). It will be apparent to those skilled in the art that thecommunication network interface 1808 can support many wired and wirelessstandards.

The optional input/output (I/O) interface 1810 is any device thatreceives input from the user and output data. The optional displayinterface 1812 is any device that may be configured to output graphicsand data to a display. In one example, the display interface 1812 is agraphics adapter.

It will be appreciated that the hardware elements of the digital device1800 are not limited to those depicted in FIG. 18. A digital device 1800may comprise more or less hardware elements than those depicted.Further, hardware elements may share functionality and still be withinvarious embodiments described herein. In one example, encoding and/ordecoding may be performed by the processor 1802 and/or a co-processorlocated on a GPU.

In various embodiments, the analysis server 208 utilizes methodologiesand/or a suite of distributed algorithms for metric/lens functioncomputations that are built on linear algebraic ideas to compute lensvalues given a data set. In some embodiments, many lens computations maybe performed as matrix-vector multiplication. After forming theappropriate linear operator, many lens computations may be performed byvector multiplication. Techniques described herein (e.g., utilizingvector multiplication in TDA analysis) may improve computationalefficiency and/or speed of analysis. Techniques described herein mayimprove scalability of larger data sets.

In one example, the L₁ Centrality lens for a data point x is defined tobe:

${{LensL}_{1}(x)} = {\sum\limits_{y}{d\left( {x,y} \right)}}$

In this equation, the sum is of distances from x to all other points yin the data set. Note that this is equivalent to finding the sum of thedistance matrix D, where Dij=d(x_(i), x_(j)), {x_(i)}=X, where X is thedata set. If the matrix-vector product D1 is computed (where “1” is thevector of all ones), the resulting vector gives the L₁ centrality forthe i^(th) point of the data set in the i^(th)entry of the vector.

In the above example, the L₁ centrality lens can be computed for a dataset by multiplying the distance matrix by a vector. The distance matrixis determined by a choice of metric, and if applied computationally“quickly” (e.g., using vector multiplication), the L₁ centrality lensmay be quickly computed. It will be appreciated that several lenses(e.g., in addition to L₁ centrality) can be applied computationally“quickly.”

In another example, the cosine distance between two points is defined tobe:d(x,y)=1−x y/∥x∥ ₂ ∥y∥ ₂An example matrix form is as follows:D _(cos)=11^(T)−diag(diag(XX^(T)))^(−1/2)XX^(T)diag(diag(XX^(T)))^(−1/2)where X is a matrix that stores the data set (m×n, where rows are datapoints). Here, again, “1” is the vector of all ones and “1^(T)” is thetranspose of the vectors of all ones. Assuming that the number of datapoints is larger than the number of data features (m>n), the aboveequation is a low-rank construction of the distance matrix.

Instead of forming the full m×m distance matrix in memory, the analysisserver 208 may perform the low-rank application, which if m>>n maysignificantly reduce the time complexity of the matrix-vectormultiplication.

There are a number of metrics and lens functions that can be computed.The following definitions and notational standards may be followed:

-   -   Data sets are denoted X, having “m” rows, and “n” columns (X is        m×n). Each row represents a data point with “n” features.    -   Distance matrices are denoted D.    -   A linear operator is a linear map on vectors O: R^(n)→R^(m). A        linear operators may be described by a matrix. Operators are not        termed as matrices herein to emphasize that while their actions        on vectors are equivalent, the actual matrix may not be formed        in memory. Examples of linear operators are data sets        (R^(m)→R^(n)), and distance matrices (R^(m)→R^(n)).    -   Multiplication by a linear operator's transpose is denoted as        adjoint multiplication.    -   “1” is used to denote the vector of all ones.    -   The Hadamard square of a matrix is another matrix with all the        entries squared. If “M”is a matrix, its Hadamard square is        “M∘M.”        Space Complexity

Space complexity refers to the memory requirements of an algorithm. Inthe example of the cosine distance matrix, storing a data set may takeO(mn) memory, and storing a vector of row two-norms takes O(m) memory.In matrix vector multiplication, O(max(m, n)) memory may be used tostore a temporary vector as the analysis server 208 successivelymultiplies it by the matrix components of the cosine distance operator.

In one example, when computing lens values for topological dataanalysis, the analysis server 208 may load a data set into ram, whichsets a baseline requirement for O(mn) memory usage. Storing a vector(O(max(m,n))) is significantly cheaper (assuming that the data set isdense, and that “n” is appreciable). As a result, the cost of storingthe cosine distance operator as described above may be dominated by thesize of the data.

In some embodiments, to make the process scalable, the analysis server208 does not store information that is quadratic in the number of datapoints. Forming a full distance matrix that requires O(m2) memory may beinfeasible for large data sets. For example, a 3 million row data setwould require 72 TB of ram to store the distance matrix with doubleprecision entries, while storing a vector of that length is only 24 MB.In some embodiments, the analysis server 208 keeps memory requirementsbelow the available RAM on a computer/cluster.

Time Complexity

Time complexity refers to the time required to perform an algorithm. Forexample, dense matrix-vector multiplication on a m×n matrix takes O(mn)time. Multiplication by a diagonal matrix can be encoded by element-wisemultiplication by two vectors. If both are length “m,” this takes O(m)time. Multiplication by a m×n rank one matrix in outer-product form maytake O(m+n) time.

As an example, consider multiplication by the cosine distance operatoras defined above: D_(cos)v, where X is m×n, and V is a m×1 vector:

$\begin{matrix}{{D_{\cos}v} = {\left( {11^{T} - {{{diag}\left( {{diag}\left( {XX}^{T} \right)} \right)}^{{- 1}/2}{XX}^{T}{{diag}\left( {{diag}\left( {XX}^{T} \right)} \right)}^{{- 1}/2}}} \right)v}} & \; \\{= {{11^{T}v} - {{{diag}\left( {{diag}\left( {XX}^{T} \right)} \right)}^{{- 1}/2}{XX}^{T}{{diag}\left( {{diag}\left( {XX}^{T} \right)} \right)}^{{- 1}/2}v}}} & \; \\{= {{11^{T}v} - {{{diag}\left( {{diag}\left( {XX}^{T} \right)} \right)}^{{- 1}/2}{XX}^{T}v_{2}}}} & {{O(m)}\mspace{14mu}{for}\mspace{14mu}{diagonal}\mspace{14mu}{scaling}} \\{= {{11^{T}v} - {{{diag}\left( {{diag}\left( {XX}^{T} \right)} \right)}^{{- 1}/2}{Xv}_{3}}}} & {{O({mn})}\mspace{14mu}{for}\mspace{14mu}{matrix}\text{-}{vector}\mspace{14mu}{mult}} \\{= {{11^{T}v} - {{{diag}\left( {{diag}\left( {XX}^{T} \right)} \right)}^{{- 1}/2}v_{4}}}} & {{O({mn})}\mspace{14mu}{for}\mspace{14mu}{matrix}\text{-}{vector}\mspace{14mu}{mult}} \\{= {{11^{T}v} - v_{5}}} & {{O(m)}\mspace{14mu}{for}\mspace{14mu}{diagonal}\mspace{14mu}{scaling}} \\{= {{1*c} - v_{5}}} & {{O(m)}\mspace{14mu}{for}\mspace{14mu}{inner}\mspace{14mu}{product}} \\{= v_{6}} & {{{O(m)}\mspace{14mu}{for}\mspace{14mu}{scaling}},{{vector}\mspace{14mu}{addition}}}\end{matrix}$

In some embodiments, the time complexity may be dominated by the twodata matrix multiplications, and takes O(mn) time. Even if the cost offorming the full distance matrix is ignored, multiplying the fulldistance matrix may take O(m²) time, which makes the above approachappealing as “m” grows larger then “n.”

FIG. 19 is a flow chart 1900 for using at least one functionrepresentable as a matrix function and more particularly, utilizingmetric matrix functions for clustering data points projected in acovered reference space for scale topological analysis. As similarlydiscussed regarding the flowchart of FIG. 8, in various embodiments, theprocessing on data and user-specified options is motivated by techniquesfrom topology and, in some embodiments, topological data analysis. Thesetechniques may be robust and general. In one example, these techniquesapply to almost any kind of data for which some qualitative idea of“closeness” or “similarity” exists. The techniques discussed herein maybe robust because the results may be relatively insensitive to noise inthe data and even to errors in the specific details of the qualitativemeasure of similarity, which, in some embodiments, may be generallyrefer to as “the distance function” or “metric.” It will be appreciatedthat while the description of the algorithms below may seem general, theimplementation of techniques described herein may apply to any level ofgenerality.

In step 1902, the input module 314 (see FIG. 3) receives data S. In oneexample, a user identifies a data structure and then identifies ID anddata fields. Data S may be based on the information within the ID anddata fields. In various embodiments, data S is treated as beingprocessed as a finite “similarity space,” where data S has a real-valuedfunction d defined on pairs of points s and t in S, such that:d(s, s)=0d(s, t)=d(t, s)d(s, t)>=0These conditions may be similar to requirements for a finite metricspace, but the conditions may be weaker. In various examples, thefunction is a metric.

It will be appreciated that data S may be a finite metric space, or ageneralization thereof, such as a graph or weighted graph. In someembodiments, data S be specified by a formula, an algorithm, or by adistance matrix which specifies explicitly every pairwise distance.

In step 1904, the input module 314 may receive a lens function andmetric function selection. The lens may be any function or combinationof functions that project data (e.g., maps data) based on data S in areference space. There may be any number of selected lens functions. Themetric function may be any function or combination of functions forclustering data in a covered reference space.

The lens and/or metric function selections may be provided by a dataanalyst, administrator, inferred from all or part of data S, in the dataS, or any other source. The lens function may be any function,including, but not limited to L1 centrality, L2 centrality, Gaussiandensity, PCA, metric PCA, MDS, or the like. In some embodiments, thelens function may utilize the metric function. The metric function maybe any metric function representable in matrix form. For example, theselected metric function may be a cosine distance between two points. Inmatrix form, this may be:D _(cos)=11^(T)−diag(diag(XX^(T)))^(−1/2)XX^(T)diag(diag(XX^(T)))^(−1/2)where again X is a matrix that stores the data set (m×n where rows aredata points). Here, again, “1” is the vector of all ones and “1^(T)” isthe transpose of the vectors of all ones. Assuming that the number ofdata points is larger than the number of data features (m>n), the aboveequation is a low-rank construction of the distance matrix.

Although the cosine distance metric is discussed herein regardingflowchart 1900, It will be appreciated that many different metricfunctions representable in matrix form may be used.

In step 1906, the input module 314 generates reference space R utilizingthe selected lens function and data S. As discussed herein, the selectedlens function may utilize the selected metric function to map data S tothe reference Space R (e.g., sec example of L₁ Centrality lens for adata point x above).

In this step, the analysis module 320 utilizes matrix vectormultiplication for application of the selected lens function using theselected metric function on al) or some of the data contained in data S.The input module 314 may store temporary vector results in O(max(m, n))memory to map the data S to the reference space R (where data S has mrows and n columns).

Reference space R may be a metric space (e.g., such as the real line).In some embodiments, the analysis module 320 generates a map ref( ) fromS into R. The map ref( ) from S into R may be called the “referencemap.” In one example, R may be Euclidean space of some dimension, but itmay also be the circle, torus, a tree, or other metric space. The mapcan be described by one or more metrics (i.e., real valued functions onS).

In various embodiments, multiplication by a m×n rank one matrix inouter-product form may take O(m+n) time.

In step 1908, the resolution module 318 generates a cover of R based onthe resolution (e.g., len(es), intervals, and overlap—see discussionregarding FIG. 7 for example). The resolution may be received from dataanalyst, administrator, inferred from all or part of data S, in the dataS, determined by outcome analysis (discussed in US Publication2016/0350389, titled “Outcome Analysis for Graph Generation,” filed May26, 2016, and incorporated herein by reference), or any other source.The cover of R may be a finite collection of open sets (in the metric ofR) such that every point in R lies in at least one of these sets. Invarious examples, R is k-dimensional Euclidean space, where k is thenumber of lens functions. More precisely in this example, R is a box ink-dimensional Euclidean space given by the product of the intervals[min_k, max_k] , where min_k is the minimum value of the k-th lensfunction on S, and max_k is the maximum value.

As discussed herein, suppose there are 2 lens functions, F1 and F2, andthat F1's values range from −1 to +1, and F2's values range from 0 to 5.Then the reference space is the rectangle in the x/y plane with corners(−1,0), (1,0), (−1, 5), (1, 5), as every point s of S will give rise toa pair (F1(s), F2(s)) that lies within that rectangle.

In various embodiments, the cover of R is given by taking products ofintervals of the covers of [min_k, max_k] for each of the k filters. Inone example, if the user requests 2 intervals and a 50% overlap for F1,the cover of the interval [−1 ,+1] will be the two intervals (−1.5,0.5), (−0.5, 1.5). If the user requests 5 intervals and a 30% overlapfor F2, then that cover of [0, 5] will be (−0.3, 1.3), (0.7, 2.3), (1.7,3.3), (2.7, 4.3), (3.7, 5.3). These intervals may give rise to a coverof the 2-dimensional box by taking all possible pairs of intervals wherethe first of the pair is chosen from the cover for F1 and the secondfrom the cover for F2. This may give rise to 2*5, or 10, open boxes thatcovered the 2-dimensional reference space. However, those skilled in theart will appreciate that the intervals may not be uniform, or that thecovers of a k-dimensional box may not be constructed by products ofintervals. In some embodiments, there are many other choices ofintervals. Further, in various embodiments, a wide range of coversand/or more general reference spaces may be used.

In one example, given a cover, C₁, . . . C_(m), of R, the reference mapis used to assign a set of indices to each point in S, which are theindices of the C_(j) such that ref(s) belongs to C_(j). This functionmay be called ref_tags(s). In a language such as Java, ref_tags would bea method that returned an int[ ]. Since the C's cover R in this example,ref(s) must lie in at least one of them, but the elements of the coverusually overlap one another, which means that points that “land near theedges” may well reside in multiple cover sets. In considering the twofilter example, if F1(s) is −0.99, and F2(s) is 0.001, then ref(s) is(−0.99, 0.001), and this lies in the cover element (−1.5, 0.5)×(−0.3,1.3). Supposing that was labeled C₁, the reference map may assign s tothe set {1}. On the other hand, if t is mapped by F1, F2 to (0.1, 2.1),then ref(t) will be in (−1.5, 0.5)×(0.7, 2.3), (−0.5, 1.5)×(0.7, 2.3),(−1.5, 0.5)×(1.7, 3.3), and (−0.5, 1.5)×(1.7, 3.3), so the set ofindices would have four elements for t.

Having computed, for each point, which “cover tags” it is assigned to,for each cover element, C_(d), the points may be constructed, whose tagsincluded, as set S(d). This may mean that every point s is in S(d) forsome d, but some points may belong to more than one such set. In someembodiments, there is, however, no requirement that each S(d) isnon-empty, and it is frequently the case that some of these sets areempty. In the non-parallelized version of some embodiments, each point xis processed in turn, and x is inserted into a hash-bucket for each j inref_tags(t) (that is, this may be how S(d) sets are computed).

It will be appreciated that the cover of the reference space R may becontrolled by the number of intervals and the overlap identified in theresolution (e.g., see further discussion regarding FIG. 7). For example,the more intervals, the finer the resolution in S—that is, the fewerpoints in each S(d), but the more similar (with respect to the lens)these points may be. The greater the overlap, the more times thatclusters in S(d) may intersect clusters in S(e)—this means that more“relationships” between points may appear, but, in some embodiments, thegreater the overlap, the more likely that accidental relationships mayappear.

In step 1910, the analysis module 320 clusters data in cover based onthe selected metric function (e.g., cosine distance) and data S (e.g.,each S(d) based on the metric function).

The significance of the user-specified inputs may now be seen. In someembodiments, the selected metric function may amount to a “forcedstretching” in a certain direction. In some embodiments, the analysismodule 320 may not cluster two points unless all of the metric values(e.g., metric values being based on data in the reference space afterapplication of the selected metric) are sufficiently “related” (recallthat while normally related may mean “close,” the cover may impose amuch more general relationship on the metric values, such as relatingtwo points s and t if ref(s) and ref(t) are sufficiently close to thesame circle in the plane).

The output may be a simplicial complex, from which one can extract its1-skeleton. The nodes of the complex may be partial clusters, (i.e.,clusters constructed from subsets of S specified as the preimages ofsets in the given covering of the reference space R).

In step 1912, the visualization engine 322 identifies nodes which areassociated with a subset of the partition elements of all of the S(d)for generating an interactive visualization. For example, suppose thatS={1,2,3,4}, and the cover is C₁, C₂, C₃. Then if ref_tags(1)={1,2, 3}and ref_tags(2)={2,3}, and ref_tags(3)={3}, and finallyref_tags(4)={1,3}, then 5(1) in this example is {1,4}, S(2)={1,2}, andS(3)={1,2,3,4}. If 1 and 2 are close enough to be clustered, and 3 and 4are, but nothing else, then the clustering for S(1) may be {1}{3}, andfor S(2) it may be {1,2}, and for S(3) it maybe {1,2}, {3,4}. So thegenerated graph has, in this example, at most four nodes, given by thesets {1}, {4}, {1,2}, and {3,4}(note that {1,2}appears in two differentclusterings). Of the sets of points that are used, two nodes intersectprovided that the associated node sets have a non-empty intersection(although this could easily be modified to allow users to require thatthe intersection is “large enough” either in absolute or relativeterms).

Nodes may be eliminated for any number of reasons. For example, a nodemay be eliminated as having too few points and/or not being connected toanything else. In some embodiments, the criteria for the elimination ofnodes (if any) may be under user control or have application-specificrequirements imposed on it. For example, if the points are consumers,for instance, clusters with too few people in area codes served by acompany could be eliminated. If a cluster was found with “enough”customers, however, this might indicate that expansion into area codesof the other consumers in the cluster could be warranted.

In step 1914, the visualization engine 322 joins clusters to identifyedges (e.g., connecting lines between nodes). Once the nodes areconstructed, the visualization engine 322 may compute intersections(e.g., edges) by computing, for each point the set of node sets. Forexample, for each s in S, node_id_set(s) may be computed, which is anint[ ]. In some embodiments, if the cover is well behaved, then thisoperation is linear in the size of the set S, and may then iterate overeach pair in node_id_set(s). There may be an edge between two node_id'sif they both belong to the same node_id_set( ) value, and the number ofpoints in the intersection is the number of different node_id sets inwhich dial pair is seen. This means that, except for the clustering step(which is often quadratic in the size of the sets S(d), but whose sizemay be controlled by the choice of cover), all of the other steps in thegraph construction algorithm may be linear in the size of S, and may becomputed quite efficiently.

In step 1916, the visualization engine 322 generates the interactivevisualization of interconnected nodes (e.g., nodes and edges displayedin FIGS. 9 and 10).

In some embodiments, in addition to computing edges (pairs of nodes),the embodiments described herein may be extended to compute triples ofnodes, etc. For example, the analysis module 320 may compute simplicialcomplexes of any dimension (by a variety of rules) on nodes, and applytechniques from homology theory to the graphs to help users understand astructure in an automatic (or semi-automatic) way.

Further, it will be appreciated that the analysis module 320 may notgenerate uniform intervals in the covering. Further, in variousembodiments, an interface may be used to encode techniques forincorporating third-party extensions to data access and displaytechniques. Farther, an interface may be used to for third-partyextensions to underlying infrastructure to allow for new methods forgenerating coverings, and defining new reference spaces.

In some embodiments, metric values (e.g., distance operators) may beconstructed from data sets using different metric functions. In someexamples, a metric function may include cosine (discussed herein),correlation, Hamming, and L.

For a correlation metric function, the correlation distance between twopoints is defined to be:d(x,y)=1−ρ(x,y)where ρ(x,y) is the Pearson correlation coefficient

${\rho\left( {x,y} \right)} = \frac{{Cov}\left( {x,y} \right)}{\sqrt{{{Var}(x)}{{Var}(y)}}}$So: d(x, y)=[x−μ((x)) ]′[y−μ((x))]/∥[x−μ(x)∥₂∥[y−μ(y)∥₂

In matrix form, this may be:D=11^(T)−diag(diag(YY^(T)))^(−1/2)YY^(T)diag(diag(YY^(T)))^(−1/2)Where Y=X−μ(X)1^(T)

For a Hamming metric function, the Hamming distance is defined to be:d(x,y)=number of different entries

In matrix form, if the data is binary, this may be:D _(hamming) =X(11^(T) −X ^(T))

For L₂, the L₂ the matrix form is:D·∘D=−2XX^(T)+diag(XX^(T))1^(T)+1diag(XX^(T))^(T)

It will be appreciated that other operators may be utilized, such as butnot limited to column-mean-centering operators, entry-wise alteredoperators, and operator-adjoint combinations.

Similarly, different lenses to map data to the reference space may beused. Different lens functions may utilize metrics in matrix operationsto achieve one or more memory and/or time efficiencies discussed herein.The following are example lens functions that may utilize at least onemetric as described herein (e.g., L₁ centrality with the cosine metric).It will be appreciated that other lens functions, combinations of lensfunctions, metric functions, and/or combinations of metric functions maybe utilized for topological data analysis than those discussed herein.

# Mat- Lens Equation Operator Vecs Exact? L₁ Centrality${L(x)} = {\sum\limits_{y}\;{d\left( {x,y} \right)}}$ D 1 yes L₂Centrality${L(x)} = \sqrt{\sum\limits_{y}\;{d\left( {x,y} \right)}^{2}}$ D/D · D 1yes Gaussian Density${L(x)} = {\exp\left( {- {\sum\limits_{y}\;{d\left( {x,y} \right)}^{2}}} \right)}$D/D · D 1 yes PCA L(x) = u₁(X) X several yes Metric PCA L(x) = u₁(D) Dseveral yes MDS L(x) = u₁(D · D) D/D · D several yes

Regarding L₁ centrality, L₁ centrality provides the L₁ norm of each rowof the distance matrix. Given a distance operator D (e.g., based on thecosign metric), the L₁ centrality of each row in the data matrix may beencoded in the vector D·1. Since the entries of D are positive, absolutevalues are not necessary.

Regarding L₂ centrality, L₂ centrality provides the L₂ norm of each rowof the distance matrix. Given a distance operator D (e.g., based on thecosign metric), the L₂ centrality of each row in the data matrix may beencoded in the vector D∘D·1 and then the entry-wise square root may betaken. In some embodiments. D∘D refers to the Hadamard square. TheHadamard square of a first matrix is a second matrix with all theentries squared.

Regarding gaussian density, similar to L₂ centrality, the Hadamardsquare of the distance operator may be utilized: D∘D·1 and an entry-wisefunction exp(−(·)) to each entry of the resulting vector.

Regarding PCA lenses (including metric PCA), using ARPACK software (theARnoldi PACKage), the top few singular values of an operator can bedetermined using a relatively small matrix multiplication. For standardPCA the data set itself is multiplied. Metric PCA requires a distanceoperator D and MDS requires the Hadamard square of the distance operatorD∘D·1.

The correlation coefficient of X and Y is ρ(X, Y)=cov(X, Y)/(std(X)std(Y) where std is the standard deviation, u₁ (X) refers to the mean ofX (e.g., E(X)), and Cov(W, Y)=E[(X−u₁(X))(Y−u₁(Y))].

FIG. 20 is a graph of an example of computing L₁ centrality over a dataset with 100 columns and a variable number of rows using vectormultiplication as discussed in some embodiments described herein. Timeis in seconds, and is averaged over 10 runs. The number of rows testedranges from 1 K to 1 M. The dotted line is the least-squares linear fit:1.65×10⁻⁷ x−1.41×10⁻³Note that there are linear asymptotics around 10K points. The leastsquares quadratic fit is:1.1×10⁻¹⁴ x ²+1.58×10⁻⁷ x−9.00×10 ⁻⁴Note that the coefficient in front of the quadratic term is close tomachine precision (i.e., asymptotic linear scaling).

In various embodiments, m>n. This is generally reasonable, as large datasets are typically large due to a large number of data points (rows),not because of a large number of features (columns). If this is nottrue, then forming the full distance matrix becomes morefeasible/attractive (unless the data is sufficiently sparse).

The above-described functions and components can be comprised ofinstructions that are stored on a storage medium (e.g., a computerreadable storage medium). The instructions can be retrieved and executedby a processor. Some examples of instructions are software, programcode, and firmware. Some examples of storage medium are memory devices,tape, disks, integrated circuits, and servers. The instructions areoperational when executed by the processor (e.g., a data processingdevice) to direct the processor to operate in accord with embodiments ofthe present invention. Those skilled in the art are familiar withinstructions, processors), and storage medium.

The present invention has been described above with reference toexemplary embodiments. It will be apparent to those skilled in the artthat various modifications may be made and other embodiments can be usedwithout departing from the broader scope of the invention. Therefore,these and other variations upon the exemplary embodiments are intendedto be covered by the present invention.

What is claimed is:
 1. A method comprising: receiving first dataassociated with data points; receiving a lens function selection, ametric function selection, and a resolution function, the metricfunction identified by the metric function selection being capable ofperforming functions on data as matrix functions; mapping second databased on the first data to a reference space by utilizing matrix vectormultiplication for application of selected lens function on second databased on the first data to map the second data to the reference space,the results of the matrix vector multiplication being stored intemporary memory enabling reduction of space complexity of memory;generating cover of reference space including the second data;clustering second data in cover based on the selected metric function todetermine each node of a plurality of nodes, each of the nodes of theplurality of nodes comprising members representative of at least onesubset of the data points; and generating a visualization comprising theplurality of nodes and a plurality of edges wherein each of the edges ofthe plurality of edges connects nodes with shared members.
 2. The methodof claim 1, wherein the metric function is a cosine distance functionrepresentable in a matrix form:D cos=11^(T)−diag(diag(XX ^(T)))^(−1/2) XX ^(T) diag(diag(XX^(T)))^(−1/2) where again X is a matrix that stores a dataset, “1” is avector of all ones and “1^(T)” is a transpose of the vectors of allones.
 3. The method of claim 1, wherein the metric function is acorrelation metric function representable in a matrix form:D=11^(T)−diag(diag(YY ^(T)))^(−1/2) YY ^(T) diag(diag(YY ^(T)))^(−1/2)where Y=X−μ1^(T).
 4. The method of claim 1, wherein the metric functionis a Hamming metric function representable in a matrix form:D _(hamming) =X(11^(T) −X ^(T))
 5. The method of claim 1, wherein themetric function is an L₂ metric function representable in a matrix form:D∘D=−2XX ^(T)+diag(XX ^(T))1^(T)+1diag(XX ^(T))^(T)
 6. The method ofclaim 1, wherein the lens function is an L₁ centrality function whereingiven a distance operator D based on the metric function, the L₁centrality of each row in a data matrix is encoded in a vector D·1. 7.The method of claim 1, wherein the lens function is an L₂ centralityfunction wherein given a distance operator D based on the metricfunction, the L₂ centrality of each row in a data matrix is encoded inan entry-wise square root of a vector D∘D·1, wherein D∘D is a Hadamardsquare.
 8. The method of claim 1, wherein the lens function is agaussian density function and wherein given a distance operator D basedon the metric function, an entry-wise function exp(−(·)) is performed ona Hadamard square of the distance operator to each entry.
 9. The methodof claim 1, wherein the lens function is a PCA function wherein thefirst data is itself multiplied.
 10. The method of claim 1, wherein thelens function is a metric PCA function that utilizes a distance operatorD based on the metric function.
 11. The method of claim 1, wherein thelens function is an MDS function that utilizes a Hadamard square of adistance operator D which is based on the metric function.
 12. Anon-transitory computer readable medium comprising instructionsexecutable by a processor to perform a method, the method comprising:receiving first data associated with data points; receiving a lensfunction selection, a metric function selection, and a resolutionfunction, the metric function identified by the metric functionselection being capable of performing functions on data as matrixfunctions; mapping second data based on the first data to a referencespace by utilizing matrix vector multiplication for application ofselected lens function on second data based on the first data to map thesecond data to the reference space, the results of the matrix vectormultiplication being stored in temporary memory enabling reduction ofspace complexity of memory; generating cover of reference spaceincluding the second data; clustering second data in cover based on theselected metric function to determine each node of a plurality of nodes,each of the nodes of the plurality of nodes comprising membersrepresentative of at least one subset of the data points; and generatinga visualization comprising the plurality of nodes and a plurality ofedges wherein each of the edges of the plurality of edges connects nodeswith shared members.
 13. The non-transitory computer readable medium ofclaim 12, wherein the metric function is a cosine distance functionrepresentable in a matrix form:Dcos=11^(T)−diag(diag(XX ^(T)))^(−1/2) XX ^(T)diag(diag(XX ^(T)))^(−1/2)where again X is a matrix that stores a dataset, “1” is a vector of allones and “1^(T)” is a transpose of the vectors of all ones.
 14. Thenon-transitory computer readable medium of claim 12, wherein the metricfunction is a correlation metric function representable in a matrixform:D=11^(T)−diag(diag(YY ^(T)))^(−1/2) YY ^(T) diag(diag(YY ^(T)))^(−1/2)where Y=X−μ1^(T).
 15. The non-transitory computer readable medium ofclaim 12, wherein the metric function is a Hamming metric functionrepresentable in a matrix form:D _(hamming) =X(11^(T) −X ^(T))
 16. The non-transitory computer readablemedium of claim 12, wherein the metric function is an L₂ metric functionrepresentable in a matrix form:D∘D=−2XX ^(T)+diag(XX ^(T))1^(T)+1diag(XX ^(T))^(T)
 17. Thenon-transitory computer readable medium of claim 12, wherein the lensfunction is an L₁ centrality function wherein given a distance operatorD based on the metric function, the L₁ centrality of each row in a datamatrix is encoded in a vector D·1.
 18. The non-transitory computerreadable medium of claim 12, wherein the lens function is an L₂centrality function wherein given a distance operator D based on themetric function, the L₂ centrality of each row in a data matrix isencoded in an entry-wise square root of a vector D∘D·1, wherein D∘D is aHadamard square.
 19. The non-transitory computer readable medium ofclaim 12, wherein the lens function is a gaussian density function andwherein given a distance operator D based on the metric function, anentry-wise function exp(−(·)) is performed on a Hadamard square of thedistance operator to each entry.
 20. The non-transitory computerreadable medium of claim 12, wherein the lens function is a PCA functionwherein the first data is itself multiplied.
 21. The non-transitorycomputer readable medium of claim 12, wherein the lens function is ametric PCA function that utilizes a distance operator D based on themetric function.
 22. The non-transitory computer readable medium ofclaim 12, wherein the lens function is an MDS function that utilizes aHadamard square of a distance operator D which is based on the metricfunction.
 23. A system comprising: one or more processors; and memorycontaining instructions executable by the processor to: receive firstdata associated with data points; receive a lens function selection, ametric function selection, and a resolution function, the metricfunction identified by the metric function selection being capable ofperforming functions on data as matrix functions; map second data basedon the first data to a reference space by utilizing matrix vectormultiplication for application of selected lens function on second databased on the first data to map the second data to the reference space,the results of the matrix vector multiplication being stored intemporary memory enabling reduction of space complexity of memory;generate cover of reference space including the second data; clustersecond data in cover based on the selected metric function to determineeach node of a plurality of nodes, each of the nodes of the plurality ofnodes comprising members representative of at least one subset of thedata points; and generate a visualization comprising the plurality ofnodes and a plurality of edges wherein each of the edges of theplurality of edges connects nodes with shared members.